Differential Equations Workbook For Dummies®

3.3. Answers to Exact First Order Differential Equation Problems

Following are the answers to the practice questions presented throughout this chapter. Each one is worked out step by step so that if you messed one up along the way, you can more easily see where you took a wrong turn.

1 Is this differential equation exact?

Answer: Yes

1. Start by casting the differential equation in this form:

   

2. So

   M(x, y) = y²

   and

   N(x, y) = 2xy

3. Now calculate the following equations:

   

   and

   

4. So

   

   Therefore, the differential equation is exact.

2 Determine whether this differential equation is exact:

Answer: No

1. Put the equation into the following form:

   

2. You know that

   M(x, y) = y

   and

   N(x, y) = xy

3. Consequently, you can calculate that

   and

   to get

   Thus, the differential equation must not be exact.

3 Following is an exact ...