If $\sin x$ is the integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$,then $P$ is

The solution of the differential equation $\frac{dy}{dx} + y \sec^2 x = \tan x \sec^2 x$ is

If the integrating factor of $x(1 - x^2)dy + (2x^2y - y - ax^3)dx = 0$ is $e^{\int Pdx}$,then $P$ is equal to

The general solution of the differential equation $x \sin x \frac{dy}{dx} + (x \cos x + \sin x)y = \sin x$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 3(\tan^2 x + 1)y = \sec^2 x$,with the initial condition $y(0) = \frac{1}{3} + e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to:

The solution of $\frac{dy}{dx} + \frac{1}{3}y = 1$ is