The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is:

The solution of the differential equation $(x + \log y)dy + y\,dx = 0$ is:

If the differential equation $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$ holds for all $x$ with initial conditions $f(0) = 1$ and $f'(0) = 2$, then which of the following is true?

If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x=2 x \sec x$ and $y(0)=0$, then which of the following is true?
$(A)$ $y(\frac{\pi}{4}) = \frac{\pi^2}{8\sqrt{2}}$
$(B)$ $y(\frac{\pi}{3}) = \frac{\pi^2}{9}$
$(C)$ $y'(\frac{\pi}{4}) = \frac{\pi^2}{8\sqrt{2}} + \frac{\pi}{\sqrt{2}}$
$(D)$ $y'(\frac{\pi}{3}) = \frac{4\pi}{3} + \frac{2\pi^2\sqrt{3}}{9}$

The solution of $(1+y^2)+(x-e^{\tan ^{-1} y}) \frac{dy}{dx}=0$ is

Let $y=y(x)$ be the solution of the differential equation $x(1-x^{2}) \frac{dy}{dx}+(3x^{2}y-y-4x^{3})=0, x>1$ with $y(2)=-2$. Then $y(3)$ is equal to