Solve y'' - 4y = 8e^(2x)

Step by step

The question

Question: solve y″ − 4y = 8e^(2x).

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 8e^(2x).

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.

But e^(2x) is already in the plain part: it gives 0 on the left.

So we multiply it by x.

Careful!

That one is already in the plain part, so it can only give 0.

Times x.

So we guess Axe^(2x) 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 = 2.

So the extra part is 2xe^(2x).

5 Add the parts

Add the parts: y = C₁e^(−2x) + C₂e^(2x) + 2xe^(2x).

6 Check

Let's check the extra part: put 2xe^(2x) into the left side.

It gives 8e^(2x): 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.