Separation of Variables/General Result

Theorem
Suppose a first order ordinary differential equation can be expressible in this form:
 * $\displaystyle g_1 \left({x}\right) h_1 \left({y}\right) + g_2 \left({x}\right) h_2 \left({y}\right) \frac {\mathrm d y} {\mathrm d x} = 0$

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

Its general solution is found by solving the integration:
 * $\displaystyle \int \frac {g_1 \left({x}\right)} {g_2 \left({x}\right)} \ \mathrm d x + \int \frac {h_2 \left({y}\right)} {h_1 \left({y}\right)} \ \mathrm d y = C$