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 $y^4+y^3+y^2=0$.

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 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?

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

Solve the differential equation $x^2{y}^{\text{'}\text{'}}-3x{y}^{\text{'}}+3y=x^4{e}^x$ using the method of variation of parameters. Which of the following is the general solution?

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

Find the general solution of the differential equation: $y^{\left(4\right)}-2y^{\left(2\right)}+y=0$.