sparkie said:
We are studying 2nd and 3rd order differential equations in class, and have touched on superposition and were talking about an equation being linearly dependent or independent. I received some good explanations from tutors about this, using vectors as examples, but I'm still a bit unclear on the concept. I'm really trying to think of a way to make the question more specific, but I don't know that I have a good enough grasp conceptually to do that. I do know that I have this one question nagging me, though:
What does linear dependence or independence mean to an nth order constant coefficient non-homogeneous ODE?
It doesn't make much sense to talk about a single equation being linearly dependent or linearly independent. It makes more sense to talk about solutions of differential equations where the solutions are linearly independent. Many of the ideas from linearly independent/dependent vectors come into play here.
With two vectors or two functions, it's easy to determine whether they are linearly independent or linearly dependent. Dependent vectors or functions are multiples of each other. With three vectors or three functions, it's not as obvious, but no one of them can be a linear combination of the others. For example, the three functions ##f_1 = \sin^2(x), f_2(x) = \cos^2(x), f_3(x) = 7## are linearly dependent, since the third function is a linear combination of the other two. Specifically, ##f_3(x) = 7f_1(x) + 7f_2(x)##.
Obtaining a set of linearly independent solutions is important to being able to specify the general solution of a differential equation, whether it's homogeneous or not. It's just a bit trickier if you're working with a nonhomogeneous diff. equation, because sometimes a solution of the homogeneous problem won't also be a solution of the nonhomogeneous problem. For example, if the diff. equation if ##y'' + y = \sin(x)##, it turns out that ##y = \sin(x)## is one solution (another is ##y = \cos(x)##) of the homogeneous problem y'' + y = 0, but it won't also be a solution of the equation ##y'' + y = \sin(x)##.
One technique that can be used to solve nonhomogeneous, linear, constant coefficient diff. equations is the method of annihilators, in which you convert a given nonhomogeneous problem into a homogeneous problem of higher order. There's an Insights article or two about this technique (written by me). You should probably be able to find them fairly easily if you search for "annihilator" among these articles.