# If the solve the problem

Question:

If $y=a\left\{x+\sqrt{x}^{2}+1\right\}^{n}+b\left\{x-\sqrt{x}^{2}+1\right\}^{-n}$, prove that $\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-n^{2}=0$

Solution:

Formula: -

(i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$

(ii) $\frac{d}{d x} x^{n}=n x^{n-1}$

(iii) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\text { wou })}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{\mathrm{dw}}{\mathrm{ds}} \cdot \frac{\mathrm{ds}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$

Given: -

$y=a\left\{x+\sqrt{x^{2}}+1\right\}^{n}+b\left\{x-\sqrt{x^{2}}+1\right\}^{-n}$

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

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{na}\left\{\mathrm{x}+\mathrm{x}^{2}+1\right\}^{\mathrm{n}}}{\sqrt{\mathrm{x}^{2}+1}}+\frac{\mathrm{nb}\left\{\mathrm{x}+\mathrm{x}^{2}+1\right\}^{-\mathrm{n}}}{\sqrt{\mathrm{x}^{2}+1}}$

$\Rightarrow \frac{\mathrm{xdy}}{\mathrm{dx}}=\frac{\mathrm{nx}}{\sqrt{\mathrm{x}^{2}+1}} \mathrm{y}$

$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{nx}}{\sqrt{\mathrm{x}^{2}+1}} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\left[\frac{\sqrt{\mathrm{x}^{2}+1}-\mathrm{x}^{2}\left(\mathrm{x}^{2}+1\right)^{-\frac{1}{2}}}{\mathrm{x}^{2}+1}\right]$

$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{n}^{2} \mathrm{x}^{2}}{\mathrm{x}^{2}+1}+\mathrm{y}\left[\frac{1}{\left(\mathrm{x}^{2}+1\right) \sqrt{\mathrm{x}^{2}+1}}\right]$

$\Rightarrow\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}=\frac{n^{2} x^{4}\left(\sqrt{x^{2}+1}\right)+x^{2} y}{x^{2}+1 \sqrt{x^{2}+1}}-\frac{n^{2} x^{2}\left(\sqrt{x^{2}+1}+y\right.}{x^{2}+1\left(\sqrt{x^{2}+1}\right)}$

Now

\begin{aligned} \Rightarrow\left(x^{2}-1\right) & \frac{d^{2} y}{d x^{2}}+\frac{x d y}{d x}-n y \\ &=\frac{n^{2} x^{4}\left(\sqrt{x^{2}+1}\right)+x^{2} y}{\left(x^{2}+1\right) \sqrt{x^{2}+1}}-\frac{n^{2} x^{2}\left(\sqrt{x^{2}+1}\right)+y}{\left(x^{2}+1\right)\left(\sqrt{x^{2}+1}\right)}-n y=0 \end{aligned}