First Order Differential Equations-Separation of Variables

If a differential equation [Maple Math] can be separated so that on one side are all [Maple Math] 's and on the other side are all [Maple Math] 's, then we can solve it by integrating both sides. For example [Maple Math] can be separated as [Maple Math] . We integrate the left hand side with respect to [Maple Math] , while we integrate the right hand side with respect to [Maple Math] . So we obtain [Maple Math] ,and thus the solution is [Maple Math] , which can be written in the explicit form as [Maple Math] . Here [Maple Math] is just another constant. We will ilustrate this method on several examples below. We also use Maple to save us great deal of work.

Example 1: Solve [Maple Math] .

Example 2: Solve [Maple Math] .

Not every first order linear equation can be solved using this method. For this method to work the equation has to be ''separable''. There is an easy way to check if the given equation is separable. It is based on the followibased on the following Theorem:

Theorem:

Example 3: Solve [Maple Math] .

[Maple Math]

Example where [Maple Math] fails:

a>