A linear first order ODE is a differential equation that is express as a linear combination of a function and it’s derivatives. It can be reduced into a Separable ODEs using an integrating factor.
The reason linear ODEs are so annoying is that they normally cannot be separated, and appear as a sum of functions instead of a product:
Solving
To solve these ODEs, the left hand side needs to be represented as a product, like , then it is separable. In order to do that, we use the product rule to find an integrating factor ().
Why?
We need such that:
Using product rule [[[Separable First Order ODEs|[Separable First Order ODEs]]]
Since we only need one factor, we can let the constant of integration, c, equal to zero.
Then, simply multiply the entire ODE by :