Step by step
The question
Question: solve dy/dx = 3x² over y², 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, 3x², 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: x³.
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 = 2.
The x side gives 0.
So C = 8/3.
4 Solve for y
Multiply both sides by 3.
Careful!
C was found with the equation it came in.
Keep it inside the cube root.
Take the cube root: y = ∛(3x³ + 8).
5 Check
Let's check with a calculator.
At x = 1, our answer gives y ≈ 2.224.
The slope between two very close points: about 0.61.
The right side at that point: about 0.61.
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.