7. Antiderivatives & Indefinite Integrals

Solve the initial value problem using the method of Laplace transforms: $y^{\mathrm{\text{'}\text{'}}}+4y=0$, $y\left(0\right)=2$, $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the following initial-value problem using Laplace transforms: $y^{\text{'}\text{'}}+4y=0$, $y\left(0\right)=2$, $y^{\text{'}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the initial-value problem: $x(x+1)\frac{dy}{dx}+xy=1$, $y\left(e\right)=1$. What is the solution $y\left(x\right)$?

Solve the initial value problem: The differential equation is homogeneous. $x{y}^2\frac{dy}{dx}=y^3-x^3$, $y\left(1\right)=4$. What is the explicit solution $y\left(x\right)$?

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