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  • Explain solution RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 52 maths

Explain solution RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 52 maths

Answers (1)

Answer:

x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0

Hint:

You must know the derivative of

        cos(log\: x)\: and\: sin(log\: x)

Given:

        \text { If } y=x^{n}\{\operatorname{acos}(\log x)+b \sin (\log x)\} \text { prove that } x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0

Solution:

        \begin{aligned} &\text { Let } y=x^{n}\{a \cos (\log x)+b \sin (\log x)\}\\ &\text { Use multiplicative rule }\\ &\mathrm{UV}=\mathrm{U}^{\prime} \mathrm{V}+\mathrm{UV}^{\prime}\\ &U=x^{n}\\ &V=a \cos (\log x)+b \sin (\log x) \end{aligned}

        \begin{aligned} &\frac{d y}{d x}=n x^{n-1}\{\operatorname{acos}(\log x)+b \sin (\log x)\}+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}=n \frac{y}{x}+\frac{x^{n}}{x}(-a \sin (\log x)+b \cos (\log x)) \\ &x \frac{d y}{d x}=n y+x^{n}(-a \sin (\log x)+b \cos (\log x)) \end{aligned}

        \begin{aligned} &\frac{d y}{d x}+x \frac{d^{2} y}{d x^{2}}=n \frac{d y}{d x}+n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}(1-x)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \end{aligned}

        \begin{aligned} &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))-\frac{1}{x} y \\ &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=\frac{n}{x}\left(x \frac{d y}{d x}-n y\right)-\frac{y}{x} \end{aligned}

        \begin{aligned} &\frac{d y}{d x}+x(1-n) \frac{d y}{d x}=n x \frac{d y}{d x}-n^{2} y-y \\ &\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(x-n x-n x)-\left(1+n^{2}\right) y=0 \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0 \end{aligned}

Posted by

Gurleen Kaur

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