Solving a linear homogenous 2nd order DE with a constant term

1. Aug 15, 2009

bitrex

I know how to go about solving differential equations of the form y''+q(x)y'+t(x)y = 0 through the methods of finding the characteristic polynomial of the differential equation and solving for the roots, etc. But what I am not clear on is how I would go about solving an equation like this where one of the terms is a constant, such as y''+q(x)y' + C = 0 or y'' +t(x)y + C = 0. I'm thinking of a situation with an F=ma equation where there is a term that is dependent on the position or velocity of an object, but there is also something like a gravitational force, which is constant.

2. Aug 15, 2009

Pengwuino

Do you mean that your q(x) and t(x) are constants? The method you described doesn't work if they are not constants. If you simply have a constant or some term that doesn't have y,y', or y'', your equation is simply inhomogeneous. You tack on a particular solution that depends on what the inhomogeneous part is. If you have a inhomogeneous part, the full solution is simply $$y(x) = y_c + y_p$$ where $$y_c$$ is the solution to the homogeneous equation and $$y_p$$ is the solution to the inhomogeneous equation. For a constant inhomogeneous part, you try $$y_p = C$$ where C is a constant. You plug this into the DE and solve for it. If the solution for your homogeneous equation is a constant, you can't use that for a particular solution. You can try $$y_p = Ax + B$$ and solve for A and B to form your solution.