Solve y'' - 2y' + y = 6

Step by step

The question

Question: solve y″ − 2y′ + y = 6.

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 6.

1 Solve with 0 on the right

Try y = e^(rx): the characteristic equation is r² − 2r + 1 = 0.

Two numbers that multiply to 1 and add to −2: −1 and −1, the same number twice.

So it is a square: (r − 1)² = 0.

Only one root: r = 1, twice.

One root gives only one solution.

The second one is the same times x.

(Put it in: it works.)

So the plain part is C₁eˣ + C₂x eˣ.

2 Guess the extra part

The extra part has the same shape as the right side: just a number.

So we guess A 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.

So the extra part is 6.

5 Add the parts

Add the parts: y = C₁eˣ + C₂x eˣ + 6.

6 Check

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

It gives 6: 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.