Step by step
The question
Question: solve dy/dx = 2y², 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, 2, 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: 2x.
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 2x.
Divide both sides by 2x + 1.
So y = −1/(2x + 1).
5 Check
Let's check with a calculator.
At x = 1, our answer gives y ≈ −0.333.
The slope between two very close points: about 0.22.
The right side at that point: about 0.22.
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.