Differential Equations Workbook For Dummies
Once you’ve figured out the type of differential equation you’re dealing with, you can move on to solving the problem by using the method of undetermined coefficients or the power series method. If a stubborn equation comes your way, try using Laplace transform solutions to help.
How to Tell One Differential Equation from Another
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:
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.
Two Effective Ways to Solve Differential Equations
You can solve a differential equation in a number of ways. The two most effective techniques you can use are the method of undetermined coefficients and the power series method.
The method of undetermined coefficients is a useful way to solve differential equations. To apply this method, simply plug a solution that uses unknown constant coefficients into the differential equation and then solve for those coefficients by using the specified initial conditions.
Power series are another tool in your differential equation solving toolkit. You can substitute a power series such as the following into a differential equation:
Then all you have to do is find a recurrence relation that gives you the coefficient an.
Solving Differential Equations Using Laplace Transform Solutions
Laplace transforms are a type of integral transform that are great for making unruly differential equations more manageable. Simply take the Laplace transform of the differential equation in question, solve that equation algebraically, and try to find the inverse transform. Here’s the Laplace transform of the function f (t):
Check out this handy table of Laplace transforms for common functions whenever you don’t want to take the time to calculate a Laplace transform on your own.