Generally, a non-homogeneous, 2nd order, linear, ordinary differential equation can be written in the form...
Equation (1) is deemed to be non-homogeneous because the term on the right-hand side of the equation is non-zero. That is...
When , equation (1) because the homogeneous ODE...
On any given open interval, let the solution of (2), the homogeneous solution be . Let be a "particular solution" of the non-homogeneous equation (1). The general solution to (1) is given by...
So, the general solution is the superposition of 2 solutions. How?
Well, to prove this, suppose we have 2 solutions of equation (1) and we name them and . Now let the difference between the solutions be...
...and let's substitute this result into (2)...
Therefore, we can conclude that the difference of 2 solutions to the non-homogeneous equation is the solution to the homogeneous equation. That is...
And rearranging a little, we have...
Now suppose that is the general solution of (1), that is...
...and is a particular solution of (1) , we have equation (3)...
As a final check, we substitute (3) into (1)...
Now, the general solution includes all solutions. From post #1, homogeneous solution is a combination of 2 linearly independent solutions...
Substituting this into (3), and the general solution is...
...where A and B are arbitrary constants of the homogeneous solution to be determined by the initial conditions.
So finally, the process in finding solutions to 2nd order, non-homogeneous ODE's is...
Find the general solution of the homogeneous equation
Find a particular solution to the non-homogeneous equation
Get it? I know, it sounds like a dog chasing its tail in that to find a solution to the non-homogeneous equation, we need to find a solution to the non-homogeneous equation.
But we'll make sense of this and go the steps to finding such solutions in the next post.
Credits:
All equations in this tutorial were created with QuickLatex
