Having found the integrating factor for a linear equation we now proceed to solve the equation.
Returning to the differential equation, written in standard form:
for which the integrating factor is
we multiply the equation by the integrating factor to obtain
At this stage the left-hand side of this equation can always be simplified as follows:
Now this is in the form of an exact differential equation and so we can integrate both sides to obtain the solution:
All that remains is to complete the integral on the right-hand side. Using the method of integration by parts we find
is the solution to the original differential equation (1). Note that, as we should expect for the solution to a first order differential equation, it contains a single arbitrary constant .
Using the integrating factors found earlier in the Task on pages 22-23, find the general solutions to the differential equations
- The standard form is for which the integrating factor is .
- The general solution is as this problem is the same as (1) with different variables.
- The equation is in standard form and the integrating factor is .