13. Intro to Differential Equations

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.

Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=17+e^x$. Which of the following is the general solution?

Which of the following functions is a solution to the differential equation $4y+y^{″}=0$?

Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=\frac{1}{3}+e^x$. Which of the following is the general solution?

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-5x}-y$ by separation of variables.

Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+5}{{8x+9}^2}$. Which of the following is the general solution?

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

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y^{\mathrm{\text{'}}}=2x+9-e^{-2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

Which of the following is the general solution to the differential equation $y^{\mathrm{\text{'}\text{'}}}+y=5x\sin \left(x\right)$ using the method of undetermined coefficients?

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?

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y=-18x^2{e}^{2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

Which of the following is the general solution to the differential equation $3x^2{y}^{\text{'}\text{'}}+6x{y}^{\text{'}}+y=0$ for $x>0$?

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-4x}-y$ by separation of variables.