Step by step
The question
Question: solve dy/dx = 4x³ over y, with y(0) = 3.
y(0) = 3 is the starting value: when x is 0, y is 3.
The right side splits into an x part, 4x³, times a y part, 1 over y.
That is a separable equation.
1 Separate x and y
Multiply 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: y² over 2.
On the right: x⁴.
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 = 3.
The x side gives 0.
So C = 9/2.
4 Solve for y
Multiply both sides by 2.
Careful!
The square root of a sum is not the sum of the roots.
It stays as one root.
Take the square root.
y(0) is positive, so we keep the plus root: y = √(2x⁴ + 9).
5 Check
Let's check with a calculator.
At x = 1, our answer gives y ≈ 3.317.
The slope between two very close points: about 1.21.
The right side at that point: about 1.21.
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.