Solve the differential equation $x\frac{dy}{dx}=6y$ by separation of variables.

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 $y\left(x\right)$?

Solve the differential equation $4y^{″}+y^{\prime }+y=x^2-5x$ using the method of undetermined coefficients. What is the general solution?

Consider the differential equation $\frac{dy}{dx}=xy+x$. Which of the following best describes this equation?

Solve the differential equation using variation of parameters: $y^{″}-16y=16x{e}^{4x}$. Which of the following is the general solution?

Solve the differential equation $3x^2{y}^{″}+7x{y}^{\prime }+y=x^2-x$ using the method of variation of parameters. Which of the following is the general solution?