Separation of Variables

Theorem
Suppose a first order ordinary differential equation can be expressible in this form:
 * $$\frac {dy} {dx} = g \left({x}\right) h \left({y}\right)$$

Then the equation is said to have separable variables, or be separable.

Its general solution is found by solving the integration:
 * $$\int \frac {dy} {h \left({y}\right)} = \int g \left({x}\right) dx + C$$

Proof
Dividing both sides by $$h \left({y}\right)$$, we get:
 * $$\frac 1 {h \left({y}\right)} \frac {dy} {dx} = g \left({x}\right)$$

Integrating both sides WRT $$x$$, we get:
 * $$\int \frac 1 {h \left({y}\right)} \frac {dy} {dx} dx = \int g \left({x}\right) dx$$

which reduces to the result.

The arbitrary constant $$C$$ happens during the integration process.