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

Step by step

The question

Question: solve dy/dx = y², with y(0) = 1.

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

The right side has only a y part, y².

The x part is just 1.

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: x.

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 x.

Divide both sides by x − 1.

So y = −1/(x − 1).

5 Check

Let's check with a calculator.

At x = 0.5, 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.