Step by step
The question
Question: solve y′ = 6xy.
The right side splits into an x part, 6x, times a y part, y.
That is a separable equation.
1 Separate x and y
Divide both sides by y, 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.
We divided by y, so y cannot be 0 here.
But y = 0 is a solution too: its slope is 0, and so is 6xy.
2 Integrate both sides
Put an integral sign on both sides.
On the left: ln |y|.
On the right: 3x².
One + C is enough: the two numbers join into one.
The integral of 1/y is the natural log of the size of y.
3 Solve for y
To undo ln, use e on both sides: e to the ln of the size of y is just |y|.
A plus in the power splits into a times: e^(3x² + C) = e^C × e^(3x²).
Careful!
The C is not added at the end.
e^C is a number that multiplies.
e^C is just some number.
With the ± from |y|, call it A: any number.
So y = Ae^(3x²).
4 Check
Let's check with a calculator.
Take A as 1, any number works.
At x = 1, our answer gives y ≈ 20.086.
The slope between two very close points: about 120.51.
The right side at that point: about 120.51.
The same slope.
Right!
Let's look back.
First: separate x and y.
Then: integrate both sides.
Then: solve for y.
Last: check.