A linear differential equation (LDE) of 1st order takes the form
where P and Q are known functions and y is the unknown. This is one of the few type of DEs that can be solved by elementary means, i.e., through algebraic operations, compositions, and integrations.
and Q be continuous on an open
interval J. Then there exists 1 and only 1 function
that satisfies both (8.1) and the initial
Using , we see that gives
is called an integrating factor since we can now integrate (
Using and , eq(8.2) is recovered. Uniqueness of this solution is guaranteed by Theorem 8.3 of §8.5.