Question: If y = (x + $\sqrt{1 + x^{2}}$)<sup>n</sup> , then (1 + x<sup>2</sup>) $\frac{d^{2}y}{dx^{2}}$ +...

Question

If y = (x + $\sqrt{1 + x^{2}}$ )ⁿ , then (1 + x²) $\frac{d^{2}y}{dx^{2}}$ + x $\frac{dy}{dx}$ is –

Answer 1

We have, y = $\left( x + \sqrt{1 + x^{2}} \right)^{n}$ ... (i)

Let $\frac{d^{2}y}{dx^{2}}$ = y₂

and $\frac{dy}{dx}$ = y₁

On differentiating Equation (i), we get

$\frac{dy}{dx}$ =n $\left\lbrack x + \sqrt{1 + x^{2}} \right\rbrack^{n - 1}$ $\left( 1 + \frac{x}{\sqrt{x^{2} + 1}} \right)$

= $\frac{n\left\lbrack x + \sqrt{1 + x^{2}} \right\rbrack^{n}}{\sqrt{1 + x^{2}}}$

Ž $\frac{dy}{dx}$ = $\frac{ny}{\sqrt{1 + x^{2}}}$

$y_{1}^{2}$ (1 + x²) = n²y²

Again differentiating, we get

2y₁y₂ (1 + x²) + 2x $y_{1}^{2}$ = 2n² yy₁

Dividing by 2y₁

y₂ (1 + x²) + xy₁ = n²y

Ž $\frac{d^{2}y}{dx^{2}}$ (1 + x²) + x $\frac{dy}{dx}$ = n² y

Question: If y = (x + \(\sqrt{1 + x^{2}}\))<sup>n</sup> , then (1 + x<sup>2</sup>) \(\frac{d^{2}y}{dx^{2}}\) +...