Solve xy' + 2y = 5x^2

Step by step

The question

Question: solve xy′ + 2y = 5x².

1 Standard form

Linear means this form: y′ + P y = Q.

P and Q have only x.

Divide everything by x, so y′ is alone at the front.

Here P = 2/x and Q = 5x.

2 The integrating factor

The integrating factor is e to the integral of P.

Multiplying by it makes the left side one derivative.

First the integral of P: the integral of 2/x is 2 ln x.

Take x > 0.

So the integrating factor is e to that power.

A number in front of ln goes up as a power: 2 ln x = ln x².

And e to the ln of something is just that something.

So the integrating factor is x².

3 Multiply both sides

Multiply both sides by x².

Every term, on both sides.

Careful!

The right side is multiplied by x² too.

Both sides, always.

4 One derivative on the left

By the product rule, the derivative of x² × y is exactly our left side.

So the left side is just (x²y)′: one derivative.

5 Integrate both sides

Integrate both sides.

The integral of a derivative is the function itself.

On the right: 5/4 × x⁴, and + C.

6 Solve for y

Divide both sides by x².

Divide each term.

x to a power over x²: take away the powers.

7 Check

Let's check: put y back in.

First y′ is 5/2 × x − 2C/x³.

Then y′ + (2/x) × y.

The C parts cancel, and we get 5x: Q again.

Right!

Let's look back.

First: standard form.

Then: the integrating factor.

Then: multiply both sides.

Then: one derivative on the left.

Then: integrate both sides.

Then: solve for y.

Last: check.