Step by step
The question
Question: solve y″ − 4y = 6e^(3x).
The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 6e^(3x).
1 Solve with 0 on the right
Try y = e^(rx): the characteristic equation is r² − 4 = 0.
Two numbers that multiply to −4 and add to 0: 2 and −2.
So it factors: (r + 2)(r − 2) = 0.
Each bracket can be 0: r = −2 or r = 2.
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^(2x).
2 Guess the extra part
The extra part has the same shape as the right side: a number times the same e power.
So we guess Ae^(3x) for the extra part.
A is 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 = 6/5.
So the extra part is 6/5 × e^(3x).
5 Add the parts
Add the parts: y = C₁e^(−2x) + C₂e^(2x) + 6/5 × e^(3x).
6 Check
Let's check the extra part: put 6/5 × e^(3x) into the left side.
It gives 6e^(3x): 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.