First Order Linear Equations
A first order linear differential equation has the
The general solution is given by
called the integrating factor. If an initial condition is given, use it to find the constant C.
Here are some practical steps to follow:
- If the differential equation is given as
rewrite it in the form
- Find the integrating factor
- Evaluate the integral
- Write down the general solution
- If you are given an IVP, use the initial condition to find
the constant C.
Example: Find the particular solution of:
Solution: Let us use the steps:
Note that you may not have to do the last step if you are asked to
find the general solution (not an IVP).
- Step 1: There is no need for rewriting the differential
equation. We have
- Step 2: Integrating factor
- Step 3: We have
- Step 4: The general solution is given by
- Step 5: In order to find the particular solution to the given
IVP, we use the initial condition to find C. Indeed, we have
Therefore the solution is
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