Solve y'' - y' = 0, y(0) = 5, y'(0) = 2

Step by step

The question

Question: solve y″ − y′ = 0, with y(0) = 5 and y′(0) = 2.

The starting values: at x = 0, y is 5 and its slope y′ is 2.

1 Try y = e to the rx

The derivatives of e to the rx are e to the rx again, times r.

So we try y = e^(rx), and look for the right r.

Its derivative: r e^(rx).

Once more: r² e^(rx).

2 The characteristic equation

Put them into the equation.

Every term has e^(rx): take it out.

e to a power is never 0.

So the bracket must be 0.

r² − r = 0: the characteristic equation.

y″ became r², y′ became r, y became 1.

3 Solve for r

Every term has an r: take it out.

A product is 0 when one part is 0: r = 0 or r = 1.

Careful!

r = 0 is a root too.

It gives e to the 0, which is 1: a plain number C₁.

4 Write the solution

Each root gives a solution e^(rx).

Any mix of the two works, so we add them with two numbers, C₁ and C₂.

So y = C₁ + C₂eˣ.

5 Use the starting values

We need y′ too: the derivative of our y.

Put x = 0 into y: C₁ + C₂ = 5.

Put x = 0 into y′: C₂ = 2.

From ①: C₁ = 5 − C₂.

Put it into ②.

Open the bracket and collect: 0 × 5 moves over.

So C₂ = 2, and C₁ = 3.

Put them in: y = 3 + 2eˣ.

6 Check

Let's check: each root must make r² − r = 0.

r = 0 gives 0.

And r = 1 gives 0.

Right!

And the starting values: y(0) = 5 and y′(0) = 2.

Right!

Let's look back.

First: try y = e to the rx.

Then: the characteristic equation.

Then: solve for r.

Then: write the solution.

Then: use the starting values.

Last: check.