Solve dy/dx = xy^2, y(0) = 1

Step by step

The question

Question: solve dy/dx = xy², with y(0) = 1.

y(0) = 1 is the starting value: when x is 0, y is 1.

The right side splits into an x part, x, times a y part, y².

That is a separable equation.

1 Separate x and y

Divide both sides by y², so y is only on the left.

Now multiply both sides by dx.

All the y parts are with dy, all the x parts with dx.

2 Integrate both sides

Put an integral sign on both sides.

On the left: −1 over y.

On the right: ½x².

One + C is enough: the two numbers join into one.

The power rule backwards, for y.

3 Use the starting value

Put in x = 0 and y = 1.

The x side gives 0.

So C = −1.

4 Solve for y

Multiply both sides by y.

Careful!

The C stays inside the bottom: it came with ½x².

Divide both sides by ½x² − 1.

No fraction inside a fraction: times the top and the bottom by 2.

So y = −2/(x² − 2).

5 Check

Let's check with a calculator.

At x = 1, our answer gives y ≈ 2.

The slope between two very close points: about 4.

The right side at that point: about 4.

The same slope.

Right!

Let's look back.

First: separate x and y.

Then: integrate both sides.

Then: use the starting value.

Then: solve for y.

Last: check.