For linear equations in the form
the solution can be found with the formula
By using integration by substitution, we get:
Proof of formula
Suppose we have a linear differential equation in the form
Let's multiply this by an integrating factor
Let's assume that the left hand side is now integratable using the reverse product rule, and then find the conditions for that make this possible.
The product rule says for two differentiable functions,
In our case, , and , and this means we require
Solving this separable differential equation is simple and yields
Going back to our earlier equation, we can now proceed and reverse the product rule
Integrating and dividing by gives us the formula above.