Step by step
The question
Question: solve dy/dx = 6x²y, with y(0) = 2.
y(0) = 2 is the starting value: when x is 0, y is 2.
The right side splits into an x part, 6x², 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.
2 Integrate both sides
Put an integral sign on both sides.
On the left: ln |y|.
On the right: 2x³.
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^(2x³ + C) = e^C × e^(2x³).
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.
4 Use the starting value
Put in x = 0 and y = 2.
The power is 0 there, and e⁰ = 1.
So A = 2.
So y = 2e^(2x³).
5 Check
Let's check with a calculator.
At x = 1, our answer gives y ≈ 14.778.
The slope between two very close points: about 88.67.
The right side at that point: about 88.67.
The same slope.
Right!
Let's look back.
First: separate x and y.
Then: integrate both sides.
Then: solve for y.
Then: use the starting value.
Last: check.