A first order differential equation is called linear if it is of the form $$\frac{dy}{dx} + P(x) y = Q(x).$$ The functions $P(x)$ and $Q(x)$ can be complicated (even non-linear!), but the left hand side is a linear function of $y$.

Example: In the video below we find the solution of the initial-value problem
$$
\frac{dy}{dx} = x + y, \qquad y(0)=0.$$

The solution of a general linear ODE follows the same steps, which we can summarize as follows:

Write the equation in standard form $$\displaystyle\frac{dy}{dx} - y = x.$$

Multiply both sides by $e^{-x}$. This is called an integrating factor. We now have $$e^{-x}\bigl(y' - y\bigr) = xe^{-x}.$$

Notice that the derivative of $ye^{-x}$ is $y'e^{-x} - ye^{-x}$ by the product rule.

Rewrite our equation as $$\left(y e^{-x}\right)' = x e^{-x}.$$

Integrate both sides. Doing the right hand side involves integration by parts. The result is $$y e^{-x} = -(x+1)e^{-x} + C,$$ or, equivalently, $$y = -(x+1) + C e^x.$$

Plug in the initial condition $y(0)=0$ to figure out $C$. Here $C=1$.