Second Order Derivative

$\mathrm{u}=\log (x^2+y^2)$ then prove $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }y\mathrm{\partial }x}$.

$y=e^x\left(a\cos x+b\sin x\right)$, prove that $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$

If $y={\tan }^{-1}x$ then prove that:

${\left(1+x^2\right)}_{y_{n+1}}+2n{x}_{y_n}+n{\left(n-1\right)}_{y_{n-1}}=0$

What is the physical significance of second order derivative ?

Find $f\text{'}\text{'}\left(x\right)$ when $f\left(x\right)=\frac{1}{x^2-4}$.

Find $f\text{'}\text{'}\left(x\right)$ when $f\left(x\right)=\frac{1}{x^2-5x+6}$.

If $y={\sin }^{-1}x$, show that $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=0$.

If $y=3e^{2x}+2e^{3x}$, prove that $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=0$.

If $y=A \sin x+B \cos x$, then prove that $\frac{d^{2}y}{d{x}^2}+y=0$.

Find $\frac{d^{2}y}{d{x}^2}$, if $y=x^3+\tan x$.

Find the $n\mathrm{th}$ derivative of $y=e^{3x}\sin x \sin \left(2x\right) \sin \left(3x\right)$.

If $y={\cos }^{-1}x$, find $\frac{d^{2}y}{d{x}^2}$ in terms of $y$ alone.

If $y={\left({\tan }^{-1}x\right)}^2$, show that 

${\left(x^2+1\right)}^2{y}_2+2x\left(x^2+1\right)y_1=2$

If $e^y\left(x+1\right)=1$ show that $y_2={y_1}^2$

If $y=5\cos x-3\sin x$, prove that $\frac{d^{2}y}{d{x}^2}+y=0$.

If $y=\sin \left(m{\sin }^{-1}x\right)$, show that $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+m^{2}y=0$.

If $y=e^{m{\cos }^{-1}x}$, show that

$\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-m^{2}y=0$

If $y=3\cos \left(\log x\right)+4\sin \left(\log x\right)$, show that $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$.

If $y=\log \left(\log 2x\right)$, show that $x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2=0$.