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Q.
If $y=\left(x+\sqrt{1+x^{2}}\right)^{n}$, then $\left(1+x^{2}\right) \frac{d^{2}\,y}{dx^{2}}+x \frac{dy}{dx}$ is
Continuity and Differentiability
Solution:
Since, $y= (x+\sqrt{1+x^{2}})^{n}$
On differentiating $w.r.t. x$, we get
$\frac{dy}{dx}=n\left(x+\sqrt{1+x^{2}}\right)^{n-1} \cdot\left(1+\frac{2x}{2\sqrt{1+x^{2}}}\right)$
$\Rightarrow \frac{dy}{dx}=\frac{n\left(x+\sqrt{1+x^{2}}\right)^{n}}{\sqrt{1+x^{2}}}$
$\Rightarrow \left(1+x^{2}\right)\left(\frac{dy}{dx}\right)^{2}=n^{2}y^{2}$
Again differentiating w.r.t. $x$, we get
$\left(1+x^{2}\right)\cdot2 \frac{dy}{dx}\cdot\frac{d^{2}\,y}{dx^{2}}+2x\left(\frac{dy}{dx}\right)^{2}$
$=n^{2}\,2\,y \frac{dy}{dx}$
$\Rightarrow \left(1+x^{2}\right) \frac{d^{2}\,y}{dx^{2}}+x \frac{dy}{dx}$
$=n^{2}\,y$