Step by step
The question
Question: solve dy/dx = −x over y.
The right side splits into an x part, −x, 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: −½x².
One + C is enough: the two numbers join into one.
The power rule backwards, for y.
3 Solve for y
Multiply both sides by 2.
2C is still just some number.
Call it C again.
Careful!
The square root of a sum is not the sum of the roots.
It stays as one root.
Take the square root: plus or minus.
So y = ±√(− x² + C).
4 Check
Let's check with a calculator.
Take C as 1, any number works.
At x = 0.5, our answer gives y ≈ 0.866.
The slope between two very close points: about −0.58.
The right side at that point: about −0.58.
The same slope.
Right!
Let's look back.
First: separate x and y.
Then: integrate both sides.
Then: solve for y.
Last: check.