Solve y'' - y = 3x^2

Step by step

The question

Question: solve y″ − y = 3x².

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 3x².

1 Solve with 0 on the right

Try y = e^(rx): the characteristic equation is r² − 1 = 0.

Two numbers that multiply to −1 and add to 0: 1 and −1.

So it factors: (r + 1)(r − 1) = 0.

Each bracket can be 0: r = −1 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^(−x) + C₂eˣ.

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 Ax² + Bx + C for the extra part.

A, B and C 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 = −3.

So B = 0.

Put in what we know.

So C = −6.

So the extra part is −3x² − 6.

5 Add the parts

Add the parts: y = C₁e^(−x) + C₂eˣ − 3x² − 6.

6 Check

Let's check the extra part: put −3x² − 6 into the left side.

It gives 3x²: 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.