Solve xy' + 3y = 10x^2, y(1) = 3

Step by step

The question

Question: solve xy′ + 3y = 10x², with y(1) = 3.

y(1) = 3 is the starting value.

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 = 3/x and Q = 10x.

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 3/x is 3 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: 3 ln x = ln x³.

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

So the integrating factor is x³.

3 Multiply both sides

Multiply both sides by x³.

Every term, on both sides.

Careful!

The right side is multiplied by x³ too.

Both sides, always.

4 One derivative on the left

By the product rule, the derivative of x³ × y is exactly our left side.

So the left side is just (x³y)′: one derivative.

5 Integrate both sides

Integrate both sides.

The integral of a derivative is the function itself.

On the right: 2x⁵, and + C.

6 Solve for y

Divide both sides by x³.

Divide each term.

x to a power over x³: take away the powers.

7 Use the starting value

Put in x = 1 and y = 3.

1 to any power is 1.

So C = 1.

So y = 2x² + 1/x³.

8 Check

Let's check: put y back in.

First y′ is 4x − 3/x⁴.

Then y′ + (3/x) × y.

The e parts cancel, and we get 10x: 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.

Then: use the starting value.

Last: check.