Solve dy/dx = -y^2

Step by step

The question

Question: solve dy/dx = −y².

The right side splits into an x part, −1, 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: −x.

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

The power rule backwards, for y.

3 Solve for y

Multiply both sides by y.

Careful!

The C stays inside the bottom: it came with −x.

Divide both sides by −x + C.

So y = −1 over (−x + C).

4 Check

Let's check with a calculator.

Take C as 1, any number works.

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: solve for y.

Last: check.