If $y=a \cos (\log x)+b \sin (\log x)$,then $x^2 y_2+x y_1=$

If $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then $\frac{d^2 y}{d x^2}=$

If $y = \sin^2 \alpha + \cos^2(\alpha + \beta) + 2 \sin \alpha \sin \beta \cos(\alpha + \beta)$,then find $\frac{d^3y}{d\alpha^3}$ (keeping $\beta$ as a constant).

If $y=A e^{m x}+B e^{n x}$,show that $\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0$.

If $y = 100 e^{2x} + 200 e^{-2x}$ and $\frac{d^2 y}{dx^2} = ay$,then $a = $ . . . . . .

$A$ non-zero polynomial with real coefficients has the property that $f''(x) f'(x) = f(x)$. Then the value of $f'''(x)$ is