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  • Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 18

Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 18

Answers (1)

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Answer:

y=\log \left(x+\sqrt{x^{2}+a^{2}}\right)^{2}  is a solution of differential equation

Hint:

Differentiate and substitute the values

Given:

y=\log \left(x+\sqrt{x^{2}+a^{2}}\right)^{2}

Solution:

Differentiating on both sides with respect to x

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left[\log \left(x+\sqrt{x^{2}+a^{2}}\right)^{2}\right] \\\\ &\frac{d y}{d x}=\frac{d}{d x}\left[2 \log \left(x+\sqrt{x^{2}+a^{2}}\right)\right] \end{aligned}

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

\begin{aligned} &\frac{d y}{d x}=\frac{2\left(x+\sqrt{x^{2}+a^{2}}\right)}{\sqrt{x^{2}+a^{2}}\left(x+\sqrt{x^{2}+a^{2}}\right)} \\\\ &\frac{d y}{d x}=\frac{2}{\sqrt{x^{2}+a^{2}}} \end{aligned}\begin{aligned} &\frac{d y}{d x}=\frac{2\left(x+\sqrt{x^{2}+a^{2}}\right)}{\sqrt{x^{2}+a^{2}}\left(x+\sqrt{x^{2}+a^{2}}\right)} \\ &\frac{d y}{d x}=\frac{2}{\sqrt{x^{2}+a^{2}}} \end{aligned}                            ...............(i)

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

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

Hence the given function is the solution to given differential equation.

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