Lesson 3, Some examples of first order linear equations.

These notes are set so that you get to prove the main results by solving smaller problems that when put together give the big result. The answers to the problems are in the videos. You will get the most out of these notes if you do (or try) the problems before looking at the videos.
A first order linear equation form $$ y' + p(x)y = f(x). $$ Our method for solving them is to multiple the equation by $$ \text{Integrating factor} = e^{\int p(x)\,dx}. $$ Then the equation becomes $$ \left( e^{\int p(x) \,dx} u\right)' = e^{\int p(x) \,dx} f(x) $$ and we can integrate to get the solution.

Problem: Find both the general solution and the solution with \(y(0)=3\) of $$ y'+2xy = 4x $$


Solution.


So far we have mostly looked a problem where we can do the integral explicitly. If we have a function \(f(x)\) were we do not know how to compute \( \int f(x)\,dx \) then we can use the Fundamental Theorem of Calculus to get an anti-derivative. That is we use that $$ \frac{d}{dx} \int_{x_0}^x f(x)\,dt = f(x). $$ The next problem, which looks like just an easy variant on the problem above, is an example where we have to use the Fundamental Theorem of Calculus to get our solution.

Problem: Find both the general solution and the solution with \(y(0)=3\) of $$ y'+2xy = 4 $$


Solution.



Problem: Here is a problem where we have to do a (very little) bit of algebra to get the equation in the form to find the integrating factor. Find both the general solution and the solution with \(u(0) = 2\) of the equation $$ ( 1+t^2) \frac{du}{dt} + 4t u = 12 t. $$


Solution.