Differential Equations Workbook For Dummies®

2.5. An Initial Peek at Separable Equations with Initial Conditions

It's a given in the world of differential equations: You're going to run into separable first order differential equations with initial conditions. Having to solve a problem with an initial condition adds another dimension to the problem, as you can see in the following example and practice problems.

NOTE

EXAMPLE

Q. Solve this differential equation:

where

y(0) = 2

A.

1. Multiply both sides of the equation by y² to get

   

2. Then multiply both sides by dx:

   y² dy = x⁴ dx

3. Integrating both sides gives you

   

4. Multiply by 3 (and absorb 3 into the constant c):

   

5. Then take the cube root:

   

6. Solve for c:

   2 = (c)^1/3

7. Here's your solution:

   

21. Figure out the answer to this differential equation:

where

y(0) = 3

22. Solve this equation: