Solve the differential equation by separation of variables: $csc\left(y\right) dx+{sec}^2\left(x\right) dy=0$. Which of the following represents the general solution?

Given the linear system of differential equations $$\begin{array}{l}\frac{\partial x}{\partial t}=2x+y\\ \frac{\partial y}{\partial t}=x+2y\end{array}$$find the most general real-valued solution.

Find the general solution of the second-order differential equation $3y^{″}+2y^{\prime }+y=0$.

Solve the differential equation by separation of variables: $\frac{dp}{dt}=p-p^2$. Which of the following is the general solution?

Solve the differential equation by variation of parameters: $y^{″}+y=\sec \left(x\right)\tan \left(x\right)$. Which of the following is the general solution?

Solve the differential equation by separation of variables: $\frac{dy}{dx}=x(1-y^2)$. Which of the following is the general solution?

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=cos\left(ex\right)$ using the method of variation of parameters. Which of the following is the general solution?

Find the general solution of the differential equation $(1+x)\frac{dy}{dx}-xy=x+x^2$.

Solve the differential equation using variation of parameters: $y^{\text{'}\text{'}}-2y^{\text{'}}+y=e^t\arctan \left(t\right)$. Which of the following is the general solution?