Suppose that $y\left(x\right)$ solves the ordinary differential equation $\frac{dy}{dx}=3y$ with the initial condition $y\left(0\right)=2$. What is $y\left(x\right)$?

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?

Solve the initial-value problem for the homogeneous differential equation: $x y^2 \frac{dy}{dx}=y^3-x^3$, with $y\left(1\right)=3$. What is the explicit solution?

Solve the differential equation $\frac{ds}{dr}=ks$ by separation of variables. Which of the following is the general solution?

Find the general solution of the differential equation $y^4+y^3+y^2=0$.

Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+7}{8x+9}$. Which of the following is the general solution?

Solve the differential equation $y^{\text{'}\text{'}}+2y^{\text{'}}=2x+7-e^{-2x}$ using the method of undetermined coefficients. Which of the following is the general solution?

Solve the following system of differential equations by systematic elimination:

$$\frac{\partial x}{\partial t}=2x-y$$$$\frac{\partial y}{\partial t}=x$$
Which of the following gives the general solution for $x\left(t\right)$?

Find the general solution of the differential equation: $x^2{y}^{\prime }+x(x+2)y=e^x$.