The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

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

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$

(A) Integrating factor is $2x$

(B) Integrating factor is $x^2$

(C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant.

(D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant.

Choose the correct answer from the options given below:

The general solution of the differential equation $(x+1)\dfrac{dy}{dx}=2e^{-y}-1$ is:

(where C is an arbitrary constant)

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

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