Separable DE: Solve $\frac{\mathrm dy}{\mathrm dx} = 2x + y$

Question

How would you solve:

$$\frac{\mathrm dy}{\mathrm dx} = 2x + y$$

I know how to solve separable equations, but I got stuck on this inseparable one. How should I approach this equation?

Answer 1

$\frac d {dx} (e^{-x}y)=e^{-x} (2x+y-y)=e^{-x}(2x)$. Integrate this equation.

Answer 2

$y'=2x+y\iff y'-y=2x\;$ is a non-homogeneous linear differential equation. The general metod here consists in

• solving the associated homogeneous equation $\;y'-y=0$,
• finding a particular solution $y_0$ of the non-homogeneous equation. As the r.h.s. is a polynomial function, you can find as a particular solution a polynomial of the same degree, i.e. $y_0=ax+b$,
• adding this particular solution to the general solution of the homogeneous equation to obtain the general solution of the non-homogeneous equation.