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: $x^2{y}^{\prime }+x(x+2)y=e^x$.

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: $y^{\left(4\right)}-2y^{\left(2\right)}+y=0$.

Find the general solution of the differential equation: ${\cos }^2\left(x\right)\sin \left(x\right)\frac{dy}{dx}+({\cos }^3(x\left)\right)y=1$.

Solve the differential equation by variation of parameters: $y^{″}+y=\sin ⁠2x$. What is the general solution $y\left(x\right)$?