Before you can solve a differential equation, you need to know what kind it is. There are several different types of equations, including linear, separable, exact, homogeneous, and nonhomogeneous.
Linear differential equations deal solely with derivatives to the first power (forget about derivatives raised to any higher power).
The power referred to here is the power the derivative is raised to, not the order of the derivative. Here’s a pretty typical-looking linear differential equation:
Separable differential equations can be written so that all terms in x and all terms in y appear on opposite sides of the equation, as you can see in this example:
which can also be written as
Exact differential equations are those where you can find a function whose partial derivatives correspond to the terms in the differential equation. Here’s an example:
Homogeneous differential equations contain only derivatives of y and terms involving y. As you can see in this equation, they’re also set to 0:
Nonhomogeneous differential equations are the same as homogeneous differential equations but with one exception: They can only have terms involving x and/or constants on the right side. Here’s an example of a nonhomogeneous differential equation:
The general solution of this nonhomogeneous differential equation:
is
where c1y1(x) + c2y2(x) is the general solution of the corresponding homogeneous differential equation
and yp(x) is a particular solution to the nonhomogeneous equation.
