Differential Equations Workbook For Dummies®

4.3. Answers to Linear Second Order Differential Equation Problems

Here are the answers to the practice questions I provide throughout this chapter. I walk you through each answer so you can see the problems worked out step by step. Enjoy!

1 Is this differential equation linear and homogeneous?

Answer: Homogeneous and linear

1. If you group all the nonconstant terms on the left side, the result equals 0, so you can cast the differential equation in this form (where the function f( ) has no constant terms):

   

   Therefore, the differential equation is homogeneous.

2. You can also put the differential equation into this form, because the exponent of y is 2:

   y″ + p(x)y′ + q(x)y = g(x)

   Consequently, the differential equation is linear.

2 Is the following equation both linear and homogeneous?

(y″)² + 4y′ + 8y = 0

Answer: Homogeneous but not linear

1. Because all nonconstant terms can be grouped on the left side of the equation, your result equals 0. Consequently, you can cast the differential equation in this form (where the function f( ) has no constant terms):

   (y″)² − f(x, y, y′) = 0

   This differential equation is homogeneous.

2. You can't put the differential equation into this form, because the exponent of y″ is 2 (not 1):

   y″ + p(x)y′ + q(x)y = g(x)

   This equation isn't linear.

3 Is this differential equation ...