Linear equations

Definition

A first order ODE is called linear if it can be written as the following

$$\frac{dy}{dt}+p(t)y=q(t)$$

Notice that linearity is enforced by $y$ term being at first power.

Notice that one can integrate the right hand side with respect to $t$ while left hand side is much trickier to do so.

So assume that there exists a $u(t)$ such that

$$\frac{d(u(t)y)}{dt}=\frac{dy}{dt}u(t)+y\frac{du}{dt}=u(t)q(t)$$

Now multiply the original equation with $u(t)$

$$u(t)\frac{dy}{dt}+u(t)p(t)y=u(t)q(t)$$

this equation holds if and only if $u(t)p(t)=du/dt$. Notice that this a separable equation for the $u$

$$\begin{array}{rl}\frac{du}{dt}& =\frac{p(t)}{\frac{1}{u}}\\ \int \frac{1}{u}du& =\int p(t)dt\\ \ln |u|& =\int p(t)dt\\ u& =\exp \left(\int p(t)dt\right)\end{array}$$

such a function $u(t)$ is called the integrating factor for the ODE.

Notice that $u(t)>0$.

Then one can see that

$$\begin{array}{rl}u(t)y& =\int u(t)q(t)dt\\ y& =\frac{\int u(t)q(t)dt}{u(t)}\end{array}$$