Step by step
The question
Question: solve dy/dx = x² over y².
The right side splits into an x part, x², times a y part, 1/y².
That is a separable equation.
1 Separate x and y
Multiply 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: y³ over 3.
On the right: ⅓x³.
One + C is enough: the two numbers join into one.
The power rule backwards, for y.
3 Solve for y
Multiply both sides by 3.
3C is still just some number.
Call it C again.
Careful!
The C stays inside the cube root: it is part of y³.
Take the cube root: y = ∛(x³ + C).
4 Check
Let's check with a calculator.
Take C as 1, any number works.
At x = 1, our answer gives y ≈ 1.26.
The slope between two very close points: about 0.63.
The right side at that point: about 0.63.
The same slope.
Right!
Let's look back.
First: separate x and y.
Then: integrate both sides.
Then: solve for y.
Last: check.