Solve y'' - 5y' + 6y = 4e^x

Step by step

The question

Question: solve y″ − 5y′ + 6y = 4eˣ.

The answer has two parts: the plain part, which solves the equation with 0 on the right, plus one extra solution that makes 4eˣ.

1 Solve with 0 on the right

Try y = e^(rx): the characteristic equation is r² − 5r + 6 = 0.

Two numbers that multiply to 6 and add to −5: −2 and −3.

So it factors: (r − 2)(r − 3) = 0.

Each bracket can be 0: r = 2 or r = 3.

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^(3x).

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 Aeˣ 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 2eˣ.

5 Add the parts

Add the parts: y = C₁e^(2x) + C₂e^(3x) + 2eˣ.

6 Check

Let's check the extra part: put 2eˣ into the left side.

It gives 4eˣ: 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.