I'm a physics student and I've just being going over some definitions for differential equations. I don't think I fully understand what a homogeneous equation is and the Wikipedia article says that a Linear DE has homogeneous solutions which add to form other homogeneous solutions. What does it mean for a solution to be homogeneous?
Usually, a DE is said to be homogeneous if it is equal to zero. eg:
$$u'' + u' -4u = 0$$
But that's not the full story is it? This definition of "homogeneous" is making me think that a homogeneous solution is equal to zero. Alas, mathematicians, please enlighten me.
Best Answer
An example of a homogeneous equation is your $$u'' -u'+4u=0.\tag{H}$$ It is a very special homogeneous equation, since it has constant coefficients.
Consider the inhomogeneous equation $$u''-u'+4u=e^t.\tag{I}$$ The article discusses, among other things, the fact that we can find all solutions of the inhomogeneous equation (I) by a) finding the general solution of the homogeneous equation (H), b) finding a single particular solution of the inhomogeneous equation (I), and c) adding the solutions found in a) and b).
Then the article focuses on finding the general solution of equations like (H), and shows that a linear combination of solutions of a homogeneous equation is always a solution of the equation.
These are solutions of the homogeneous equation, and not "homogeneous solutions." In the context of linear differential equations, homogeneous solution has no meaning.