Calculus 2: Linear differential equations

Find the functions where the second derivative of the function plus two times the first derivative of the function is equal to three times the function itself.

Find the general solution of the first-order linear differential equation $\displaystyle \frac{dy}{dx} + \frac{2}{x} y = 3x - 5$.

Solve a first order linear ordinary differential equation using the integrating factor method.

Solve the differential equation using the method of integrating factors, where the initially given differential equation is linear with the coefficient functions for $y$ and $y'$ dependent on $x$.

Given that $y(0)=\frac{4}{3}$, find the specific solution for $y(x)$ using the method of integrating factors.

Solve the differential equation: $\frac{dy}{dx} + 5y = e^{2x}$.

Solve the differential equation: $\frac{dy}{dx} - y = x$.