Question

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

Disclaimer: There is a misprint in the question, $\left(x^2+1\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-n^{2}y=0$ must be written instead of $\left(x^2-1\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-n^{2}y=0.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ y=a{\left\{x+\sqrt{x^2+1}\right\}}^n+b{\left\{x-\sqrt{x^2+1}\right\}}^{-n}...\left(1\right) \\ \mathrm{Differentiating}y\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ \frac{dy}{dx}=an{\left\{x+\sqrt{x^2+1}\right\}}^{n-1}\left(1+\frac{1}{2\sqrt{x^2+1}}\times 2x\right)-bn{\left\{x-\sqrt{x^2+1}\right\}}^{-n-1}\left(1-\frac{1}{2\sqrt{x^2+1}}\times 2x\right) \\ =an{\left\{x+\sqrt{x^2+1}\right\}}^{n-1}\left(1+\frac{x}{\sqrt{x^2+1}}\right)-bn{\left\{x-\sqrt{x^2+1}\right\}}^{-n-1}\left(1-\frac{x}{\sqrt{x^2+1}}\right) \\ =an{\left\{x+\sqrt{x^2+1}\right\}}^{n-1}\left(\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}}\right)-bn{\left\{x-\sqrt{x^2+1}\right\}}^{-n-1}\left(\frac{\sqrt{x^2+1}-x}{\sqrt{x^2+1}}\right) \\ =an{\left\{x+\sqrt{x^2+1}\right\}}^{n-1}\left(\frac{x+\sqrt{x^2+1}}{\sqrt{x^2+1}}\right)+bn{\left\{x-\sqrt{x^2+1}\right\}}^{-n-1}\left(\frac{x-\sqrt{x^2+1}}{\sqrt{x^2+1}}\right) \\ =\left\{a{\left\{x+\sqrt{x^2+1}\right\}}^n\left(\frac{n}{\sqrt{x^2+1}}\right)+b{\left\{x-\sqrt{x^2+1}\right\}}^{-n}\right\}\left(\frac{n}{\sqrt{x^2+1}}\right) \\ =\left(\frac{n}{\sqrt{x^2+1}}\right)y\left[\mathrm{From}\left(1\right)\right] \\ \Rightarrow \sqrt{x^2+1}\frac{dy}{dx}=ny \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \left(x^2+1\right){\left(\frac{dy}{dx}\right)}^2=n^2{y}^2...\left(2\right) \\ \mathrm{Differentiating}\left(2\right)\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ \left(x^2+1\right)2\frac{dy}{dx}\times \frac{d^{2}y}{d{x}^2}+2x{\left(\frac{dy}{dx}\right)}^2=n^2\left(2y\frac{dy}{dx}\right) \\ \Rightarrow \left(x^2+1\right)\frac{d^{2}y}{d{x}^2}+x\left(\frac{dy}{dx}\right)=n^2\left(y\right) \\ \Rightarrow \left(x^2+1\right)\frac{d^{2}y}{d{x}^2}+x\left(\frac{dy}{dx}\right)-n^{2}y=0 \\ \mathrm{Hence},\left(x^2+1\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-n^{2}y=0. \end{gathered}$$