Step by step
The question
Question: solve y″ + 3y′ + 2y = 4x.
The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 4x.
1 Solve with 0 on the right
Try y = e^(rx): the characteristic equation is r² + 3r + 2 = 0.
Two numbers that multiply to 2 and add to 3: 2 and 1.
So it factors: (r + 2)(r + 1) = 0.
Each bracket can be 0: r = −2 or r = −1.
Each root gives a solution e^(rx).
Any mix of the two works, so we add them with two numbers, C₁ and C₂.
So the plain part is C₁e^(−2x) + C₂e^(−x).
2 Guess the extra part
The extra part has the same shape as the right side: a polynomial of the same size, with every lower power too.
Careful!
Every lower power is needed too: the derivatives make them.
So we guess Ax + B for the extra part.
A and B are the numbers we must find.
3 Put the guess in
Its derivative.
And once more.
Put them into the left side.
Multiply out and put like parts together.
4 Match the parts
The two sides must match, part by part: the same amount of each shape.
So A = 2.
Put in what we know.
So B = −3.
So the extra part is 2x − 3.
5 Add the parts
Add the parts: y = C₁e^(−2x) + C₂e^(−x) + 2x − 3.
6 Check
Let's check the extra part: put 2x − 3 into the left side.
It gives 4x: the right side.
Right!
Let's look back.
First: solve with 0 on the right.
Then: guess the extra part.
Then: put the guess in.
Then: match the parts.
Then: add the parts.
Last: check.