General Solutions

The general solution to a differential equation is the function such that the differential equation is true. Where , and are arbitrability defined (depending on the problem).

Breakdown - Linear only?

First lets observe this general solution:

The solution is made up of two components, a complementary solution and a particular solution.

The complementary solution is the solution if the original differential equation was homogeneous, but if the equation isn't then we need to account for that. This is what the particular solution is for.

This means that if our original differential equation was in fact homogenous then the complimentary solution is the general solution!

For Scalar Equations

For Linear Systems of Equations

We find that when working with nonhomogeneous linear systems the general solution takes the form

This is because the complimentary solution is actually just the solution to the linear system of differential equations if it was homogeneous (if it was equal to zero). The particular solution is then used to account for the fact that the system is not homogeneous we use