Differential Equations Workbook For Dummies®

Chapter 7. Taking On Nonhomogeneous Linear Higher Order Differential Equations

In This Chapter

• Finding answers with the help of Ae^rx

• Working with polynomial differential equations

• Knowing what to do when you see sines and cosines

In this chapter, you work with general nonhomogeneous linear higher order differential equations (which are sometimes referred to as nth order equations) that look like this:

Such an equation may seem complex, but you can easily solve it by using the method of undetermined coefficients for nonhomogeneous higher order differential equations.

NOTE

The method of undetermined coefficients says that if g(x) has a certain form, then you must attempt to find a particular solution of a similar form. After you find the particular solution, you must solve for the general solution, which is the sum of the homogeneous solution (which you find by setting g(x) to 0) and the particular solution.

The various forms of g(x) give you a clue as to what the form of the particular solution may be. If g(x) equals

• e^rx, then try a particular solution of the form Ae^rx, where A is a constant. Because derivatives of e^rx reproduce e^rx, you have a good chance of finding a particular solution this way.

• a polynomial of order n, then try a polynomial of order n.

• a combination of sines and cosines, sin αx + cos βx, then try a combination of sines and cosines with undetermined coefficients ...