Differential Equations Workbook For Dummies®

Chapter 5. Tackling Nonhomogeneous Linear Second Order Differential Equations

In This Chapter

• Refreshing your memory of the method of undetermined coefficients

• Working with g(x) in its various forms

Welcome to the wonderful world of nonhomogeneous second order differential equations! (If you're thinking "Homoge-huh?" flip to Chapter 4 for a refresher on what makes a second order differential equation homogeneous in the first place.) In other words, here's your chance to play with equations that look like this:

y″ + p(x)y′ + q(x)y = g(x)

where g(x) ≠ 0.

You, lucky person that you are, get to handle linear second order differential equations like this one in the following pages:

y″ − y′ − 2y = 10e⁴x

NOTE

The method of undetermined coefficients advises that when you find a candidate solution, y, and plug it into the left-hand side of the equation, you end up with g(x). Because g(x) is just a function of x, you can often guess the form of y_p(x), up to arbitrary coefficients, and then solve for those coefficients by plugging y_p(x) into the differential equation.

This method works because you're handling only g(x), and the form of g(x) can often tell you what a particular solution looks like. For example, if g(x) is in the form of erx, then try a particular solution of the form Aerx, where A is a constant. Because derivatives of erx reproduce erx, you have a good chance of finding a ...