A separable differential equation is an equation of the form

,

We can not integrate the equation immediately because the right-hand side contains an unknown function  y (x) together with the function of independent variable  x.

In order to integrate this equation, it is necessary, first, to separate the factors depending on different variables, x and y:

.

Then integrate both sides to obtain the general integral of the equation:

.

A differential equation of the form

can be reduced to a separable equation by substitution  u (x) :

Then the general integral can be found by integration:

Finally, we have to solve the ordinary linear equation

u = ax + by + c

with respect to  y,

,

substituting the result of integration for u.

Indefinite Integrals Definite Integrals Differential Equations