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It represents physical rates of change, such as velocity or growth rates, and is written as ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.415em", verticalAlign: "-0.345em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "y" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: 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Yeh physics aur engineering ke dynamics (rates of change) ko represent karne ke liye use hoti hai."] }) })] }), "\n", _jsxs(Explanation, { children: [_jsxs(English, { children: [_jsx(_components.h3, { children: "Order and Degree of a Differential Equation" }), _jsx(_components.p, { children: "Every differential equation is characterized by two key attributes:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Order:" }), " The order of a differential equation is the order of the highest derivative appearing in it."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["For example, in ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: 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"-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] }) }), ", the highest derivative is the second derivative, so the ", _jsx(_components.strong, { children: "Order is 2" }), "."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Degree:" }), " The degree of a differential equation is the power/exponent of the highest order derivative, provided the equation is written as a polynomial in 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[_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.07em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7463em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: 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}), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.004em", verticalAlign: "-0.65em" } }), _jsxs(_components.span, { className: "minner", children: [_jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size2", children: "(" }) }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9322em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size2", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.354em" }, children: _jsxs(_components.span, { style: { top: "-3.6029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "4" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] }) }), ", the highest derivative is ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.415em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.07em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7463em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) }) }), " (order 2) and its power is 3, so the ", _jsx(_components.strong, { children: "Degree is 3" }), "."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "Note:" }), " If a derivative is inside a trigonometric, exponential, or logarithmic function (e.g., ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2772em", verticalAlign: "-0.345em" } }), _jsx(_components.span, { className: "mop", children: "sin" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9322em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), " or ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.888em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.888em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" }), _jsx(_components.span, { className: "mord mtight", children: "/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }) }) }) }) })] })] }) }) }), "), the degree is ", _jsx(_components.strong, { children: "undefined" }), "."] }), "\n"] }), "\n"] }), "\n"] }), _jsx(_components.h3, { children: "Formation of a Differential Equation" }), _jsx(_components.p, { children: "The general equation of a family of curves contains arbitrary constants. To form the corresponding differential equation:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["Count the number of arbitrary constants (say, ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), ")."] }), "\n", _jsxs(_components.li, { children: ["Differentiate the curve's equation ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), " times."] }), "\n", _jsx(_components.li, { children: "Eliminate the arbitrary constants using the original equation and the derived equations." }), "\n", _jsxs(_components.li, { children: ["The resulting equation containing derivatives is the required differential equation. Its order will be exactly ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), "."] }), "\n"] })] }), _jsxs(Hinglish, { children: [_jsx(_components.h3, { children: "Order aur Degree Kya Hote Hain?" }), _jsx(_components.p, { children: "Differential equation ki do main properties hoti hain:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Order:" }), " Differential equation mein present sabse bade (highest) derivative ke number ko Order kehte hain."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Jaise ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.415em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.07em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7463em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2772em", verticalAlign: "-0.345em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9322em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] }) }), " mein sabse bada derivative second derivative hai, toh iska ", _jsx(_components.strong, { children: "Order 2" }), " hai."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Degree:" }), " Highest derivative ki power ko Degree kehte hain, lekin equation roots (fractional powers) se free honi chahiye."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Jaise ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.004em", verticalAlign: "-0.65em" } }), _jsxs(_components.span, { className: "minner", children: [_jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size2", children: "(" }) }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.07em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7463em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size2", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { 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className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9322em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] }) }), " mein highest derivative ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.415em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.07em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7463em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) }) }), " hai aur uski power 3 hai, toh iska ", _jsx(_components.strong, { children: "Degree 3" }), " hai."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "Note:" }), " Agar derivative kisi log, sin, cos ya exponential ke andar phansa ho (e.g. ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2772em", verticalAlign: "-0.345em" } }), _jsx(_components.span, { className: "mop", children: "sin" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9322em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "x" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), "), toh degree ", _jsx(_components.strong, { children: "not defined" }), " hoti hai."] }), "\n"] }), "\n"] }), "\n"] }), _jsx(_components.h3, { children: "Formation of a Differential Equation (DE Kaise Banayein?)" }), _jsx(_components.p, { children: "Kisi curve ki equation se differential equation banane ke steps:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["Equation mein arbitrary constants (jaise ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" })] }) }) }), ") count karein (let's say count ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), " hai)."] }), "\n", _jsxs(_components.li, { children: ["Equation ko ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), " baar differentiate karein."] }), "\n", _jsx(_components.li, { children: "Purani aur nayi equations ko use karke arbitrary constants ko remove (eliminate) karein." }), "\n", _jsxs(_components.li, { children: ["Constants remove hone ke baad jo final derivatives wali equation bachegi, wahi required DE hogi. Iska order ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) }) }), " hi hoga."] }), "\n"] })] }), _jsx(ComparisonTable, { children: _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsx(_components.th, { style: { textAlign: "left" }, children: "Differential Equation" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Highest Derivative" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Order" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Exponent of Highest Derivative" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Degree" })] }) }), 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"mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "y" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: "2" }), _jsx(_components.td, { style: { textAlign: "left" }, children: "Inside trigonometric term" }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.strong, { children: "Not Defined" }) })] })] })] }) })] }), "\n", _jsxs(Summary, { children: [_jsx(English, { children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The ", _jsx(_components.strong, { children: "Order" }), " is the highest derivative count; the ", _jsx(_components.strong, { children: "Degree" }), " is the exponent of that highest derivative (must clear fractional exponents first)."] }), "\n", _jsx(_components.li, { children: "Degree is undefined if derivatives are arguments of non-algebraic functions (trig, log, exp)." }), "\n", _jsx(_components.li, { children: "Number of arbitrary constants in a curve's equation determines the order of the resulting differential equation." }), "\n"] }) }), _jsx(Hinglish, { children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Order" }), " highest derivative ka level hota hai; ", _jsx(_components.strong, { children: "Degree" }), " us highest derivative ki power hoti hai (pehle roots/fractions hatana zaroori hai)."] }), "\n", _jsxs(_components.li, { children: ["Agar derivatives non-algebraic functions ke andar hon (jaise ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0019em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mop", children: "cos" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), "), toh degree not defined hoti hai."] }), "\n", _jsx(_components.li, { children: "Kisi curve ki equation mein jitne constants honge, uske differential equation ka order utna hi hoga." }), "\n"] }) })] }), "\n", _jsx(ImportantQA, { questions: [{ "q": "Determine the order and degree of the differential equation: (d^2y/dx^2)^2 = sqrt(1 + dy/dx).", "marks": 2, "a": { "components": [{ "type": "Explanation", "content": "Given equation: $\\left(\\frac{d^2y}{dx^2}\\right)^2 = \\sqrt{1 + \\frac{dy}{dx}}$.\nTo find the degree, we must first make the equation free from radicals. Squaring both sides:\n$$\\left[\\left(\\frac{d^2y}{dx^2}\\right)^2\\right]^2 = 1 + \\frac{dy}{dx}$$\n$$\\left(\\frac{d^2y}{dx^2}\\right)^4 = 1 + \\frac{dy}{dx}$$\nNow, the highest derivative is $\\frac{d^2y}{dx^2}$, which has a power of 4.\nTherefore:\n- **Order = 2**\n- **Degree = 4**.Diya gaya equation: $\\left(\\frac{d^2y}{dx^2}\\right)^2 = \\sqrt{1 + \\frac{dy}{dx}}$.\nDegree nikalne ke liye hume radical sign (root) hatana padega. Dono sides square karne par:\n$$\\left(\\frac{d^2y}{dx^2}\\right)^4 = 1 + \\frac{dy}{dx}$$\nAb highest derivative $\\frac{d^2y}{dx^2}$ hai aur uski power 4 hai.\nIsliye:\n- **Order = 2**\n- **Degree = 4**." }] } }, { "q": "Form the differential equation for the curve y = A * e^(2x) + B * e^(-2x), where A and B are arbitrary constants.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given: $y = A e^{2x} + B e^{-2x}$ --- (Eq. 1).\nSince there are two arbitrary constants ($A$ and $B$), we must differentiate twice.\nDifferentiating once with respect to $x$:\n$\\frac{dy}{dx} = 2A e^{2x} - 2B e^{-2x}$ --- (Eq. 2).\nDifferentiating again with respect to $x$:\n$\\frac{d^2y}{dx^2} = 4A e^{2x} + 4B e^{-2x}$\n$\\frac{d^2y}{dx^2} = 4(A e^{2x} + B e^{-2x})$.\nFrom Eq. 1, we know $y = A e^{2x} + B e^{-2x}$. Substituting this:\n$\\frac{d^2y}{dx^2} = 4y$\n$\\frac{d^2y}{dx^2} - 4y = 0$.\nThis is the required differential equation.Diya hai: $y = A e^{2x} + B e^{-2x}$ --- (Eq. 1).\nKyunki do constants ($A$ aur $B$) hain, isliye do baar differentiate karenge.\nFirst derivative: \n$\\frac{dy}{dx} = 2A e^{2x} - 2B e^{-2x}$ --- (Eq. 2).\nSecond derivative:\n$\\frac{d^2y}{dx^2} = 4A e^{2x} + 4B e^{-2x}$\n$\\frac{d^2y}{dx^2} = 4(A e^{2x} + B e^{-2x})$.\nEq. 1 se hume pata hai $y = A e^{2x} + B e^{-2x}$. Value put karne par:\n$\\frac{d^2y}{dx^2} = 4y$\n$\\frac{d^2y}{dx^2} - 4y = 0$.\nYeh required differential equation hai." }] } }, { "q": "Form the differential equation of the family of curves y^2 = 4*a*x, where a is the parameter.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given curve: $y^2 = 4ax$ --- (Eq. 1).\nThere is only one arbitrary constant $a$, so we differentiate once.\nDifferentiating with respect to $x$:\n$2y \\frac{dy}{dx} = 4a$ --- (Eq. 2).\nTo eliminate $4a$, substitute $4a = 2y \\frac{dy}{dx}$ from Eq. 2 into Eq. 1:\n$y^2 = \\left(2y \\frac{dy}{dx}\\right) x$\n$y^2 = 2xy \\frac{dy}{dx}$.\nDividing both sides by $y$ (assuming $y \\neq 0$):\n$y = 2x \\frac{dy}{dx}$\n$2x \\frac{dy}{dx} - y = 0$.\nThis is the required differential equation.Diya gaya curve: $y^2 = 4ax$ --- (Eq. 1).\nIsmein ek constant $a$ hai, toh hum ek baar differentiate karenge.\nDifferentiate karne par:\n$2y \\frac{dy}{dx} = 4a$ --- (Eq. 2).\n$4a$ ko remove karne ke liye, Eq. 2 se $4a$ ki value Eq. 1 mein substitute karein:\n$y^2 = \\left(2y \\frac{dy}{dx}\\right) x$\n$y^2 = 2xy \\frac{dy}{dx}$.\nDono sides ko $y$ se divide karne par:\n$y = 2x \\frac{dy}{dx}$\n$2x \\frac{dy}{dx} - y = 0$.\nYeh required differential equation hai." }] } }] }), "\n", _jsx(MCQ, { questions: [{ "id": "cb7a9f9a-e41d-4d84-bcf8-9e5b969712ad", "q": "What is the order of the differential equation: d^3y/dx^3 + (dy/dx)^5 + y = 0?", "options": ["1", "5", "3", "0"], "answer": 2, "hint": "Order is defined by the highest derivative present in the equation.", "explanation": "The highest derivative present is the third derivative (d^3y/dx^3), so the order is 3. The power of 5 is on the first derivative, which does not affect the order." }, { "id": "55f12007-a73e-47a9-acc8-fd1e107849f4", "q": "Find the degree of the differential equation: d^2y/dx^2 + (dy/dx)^3 + y^2 = 0.", "options": ["1", "2", "3", "Undefined"], "answer": 0, "hint": "Look at the exponent of the highest derivative (d^2y/dx^2).", "explanation": "The highest derivative is d^2y/dx^2. Its exponent (power) is 1. Thus, the degree is 1." }, { "id": "22b0c286-820e-4b61-8e91-e6b87232b16d", "q": "What is the degree of: d^2y/dx^2 + sin(dy/dx) = 0?", "options": ["1", "2", "Not defined", "0"], "answer": 2, "hint": "Notice that the derivative term dy/dx is an argument inside the sin function.", "explanation": "Since the derivative dy/dx is inside the trigonometric sine function, the equation cannot be written as a polynomial in derivatives, so the degree is not defined." }, { "id": "03e09545-7ea5-486d-bd2f-9d5b70c57d5f", "q": "How many arbitrary constants are present in the general solution of a 3rd-order differential equation?", "options": ["1", "2", "3", "Depends on the function"], "answer": 2, "hint": "The order of a differential equation equals the number of arbitrary constants in its general solution.", "explanation": "A differential equation of order n will have exactly n arbitrary constants in its general solution. Hence, for order 3, there are 3 constants." }, { "id": "e70ff2a3-b1c2-4d22-9306-7ee9eeb98663", "q": "Form the differential equation for y = c * x where c is an arbitrary constant.", "options": ["dy/dx = y/x", "dy/dx = x/y", "dy/dx = c", "x * dy/dx + y = 0"], "answer": 0, "hint": "Differentiate once to get dy/dx = c, then substitute c = y/x.", "explanation": "y = cx => dy/dx = c. Substitute c = y/x into the derivative equation: dy/dx = y/x." }, { "id": "955181b3-55cd-4b3c-87a1-bdba7204a358", "q": "Find the order and degree of: sqrt(d^2y/dx^2) = dy/dx + 1.", "options": ["Order 2, Degree 1", "Order 2, Degree 1/2", "Order 2, Degree 2", "Order 1, Degree 2"], "answer": 0, "hint": "Square both sides to remove the square root, then determine order and degree.", "explanation": "Squaring both sides gives d^2y/dx^2 = (dy/dx + 1)^2. The highest derivative is d^2y/dx^2 (order 2), and its power is now 1. Thus, Order is 2 and Degree is 1." }, { "id": "5b1be240-794d-49c3-aaaf-8eccdc4af417", "q": "The order of the differential equation of all circles of constant radius 'r' is:", "options": ["1", "2", "3", "4"], "answer": 1, "hint": "The general equation of a circle of radius r is (x-h)^2 + (y-k)^2 = r^2. How many parameters are there?", "explanation": "The circle equation has two arbitrary constants, h and k (the coordinates of the center). The radius r is constant. Thus, there are 2 parameters, giving an order of 2." }, { "id": "bdd6db43-2adc-4940-b4b7-94cea68999bc", "q": "Form the differential equation of y = A * cos(x) + B * sin(x) by eliminating A and B.", "options": ["d^2y/dx^2 + y = 0", "d^2y/dx^2 - y = 0", "dy/dx + y = 0", "d^2y/dx^2 = 0"], "answer": 0, "hint": "Differentiate twice. You will find that d^2y/dx^2 = -y.", "explanation": "y = A cos x + B sin x => y' = -A sin x + B cos x => y'' = -A cos x - B sin x = -(A cos x + B sin x) = -y. So, y'' + y = 0." }, { "id": "cbd602ce-0472-480f-a4b7-26fe3c84a954", "q": "What is the degree of the differential equation: (d^2y/dx^2)^3 + log(d^2y/dx^2) = 0?", "options": ["3", "1", "Not defined", "2"], "answer": 2, "hint": "Look at the log function. Is a derivative nested inside it?", "explanation": "Since a derivative term (d^2y/dx^2) is inside the logarithmic function, the equation cannot be expressed as a polynomial in derivatives. Therefore, the degree is not defined." }, { "id": "3e1bb109-296b-4642-9390-b6410aa58d06", "q": "Form the differential equation for the family of curves y = A * e^(3x), where A is an arbitrary constant.", "options": ["dy/dx = 3y", "dy/dx = y", "dy/dx = 3A", "d^2y/dx^2 = 9y"], "answer": 0, "hint": "Differentiate once to get dy/dx = 3 * A * e^(3x), then substitute y = A * e^(3x).", "explanation": "y = A*e^(3x) => dy/dx = 3*A*e^(3x) = 3y. Hence, dy/dx = 3y." }], topicId: "3.1" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent }; function _missingMdxReference(id, component) { throw new Error("Expected " + (component ? "component" : "object") + " `" + id + "` to be defined: you likely forgot to import, pass, or provide it."); }