In this Section we consider differential equations which can be written in the form
Note that the right-hand side is a product of a function of , and a function of . Examples of such equations are
Not all first order equations can be written in this form. For example, it is not possible to rewrite the equation
in the form
Determine which of the following differential equations can be written in the form
If possible, rewrite each equation in this form.
- cannot be written in the stated form,
- Reformulating gives which is in the required form.
The variables involved in differential equations need not be and . Any symbols for variables may be used. Other first order differential equations are
Given a differential equation in the form
we can divide through by to obtain
If we now integrate both sides of this equation with respect to we obtain
We have separated the variables because the left-hand side contains only the variable , and the right-hand side contains only the variable . We can now try to integrate each side separately. If we can actually perform the required integrations we will obtain a relationship between and . Examples of this process are given in the next subsection.
Method of Separation of Variables
The solution of the equation