Math 3321 – Lecture 4 notes
Today we will look at solving:
First Order Linear Differential Equations
dy
+ p(x)y = f (x)
dx
Examples:
How to solve a linear differential equation y′ + p(x)y = f (x)
1. Make the left side look like the derivative of a product by multiplying both sides
by a special function called the integrating factor.
d
recall product rule: dx ( u(x)⋅ v(x)) = u(x)⋅ v′(x) + u ′(x)⋅ v(x)
2. Let u(x) be your integrating factor, then you will have u(x) ( y′ + p(x)y ) = f (x)u(x)
So, u(x) y′ + u(x)p(x)y = f (x)u(x) => we need u(x)p(x) to equal u ′(x) (note that this
is a separable diff eq.
Let’s look at u ′(x) = u(x)p(x)
3. Now we have
Integrate…