Differential Equations Workbook For Dummies®

4.1. Getting the Goods on Linear Second Order Differential Equations

NOTE

In linear second order differential equations, the exponent of y″, y′, and y is 1. Equations not in that form are called nonlinear (but don't worry, I don't deal with those nasty things here).

The equation I present at the beginning of this chapter is the typical form used for linear second order differential equations in most textbooks. However, some textbooks seem to just want to make your life difficult, so they write the equation as follows:

P(x)y″ + Q(x)y′ + R(x)y = G(x)

Note that the only difference in this form is that y″ has a coefficient, P(x).

You can convert such equations into the first format I present simply by noting that and and

Of course, linear second order differential equations can also be homogeneous, meaning that in an equation such as the following, g(x) = 0:

y″ + p(x)y′ + q(x)y = 0

Using the P(x), Q(x), R(x), and G(x) terminology that some textbooks prefer, note that you can also rewrite this equation with G(x) = 0:

P(x)y″ + Q(x)y′ + R(x)y = 0

If a linear second order differential ...