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Unformatted text preview: 18.03 Lecture #36 Dec. 4, 2009: notes So far we’ve said a lot about how to solve the homogeneous system x ′ ( t ) = A x ( t ) . (Homogeneous system ODE) Just as in the one-dependent-variable case, homogeneous systems often describe the behavior of physical systems (electrical networks, economic systems, populations . . . ) when nothing external is acting. Of course we are very often interested in the response of the system to external influences. In this setting the differential equation often takes the form x ′ ( t ) = A x ( t ) + f ( t ) . (Inhomogeneous system ODE) Here the i th coordinate of f represents the external action on the i th unknown function. As usual we will sometimes care about initial conditions x ( t ) = x . (Initial conditions) The theoretical basis for solving these equations comes immediately from the version of the Super- position Principle that appeared in Lecture 33: Inhomogeneous Superposition Theorem. Suppose x 1 , x 2 , . . . x n are independent solu- tions of (Homogeneous system ODE), so that the linear combinations x c = c 1 x 1 + c 2 x 2 + ··· + c n x n are all possible solutions of (Homogeneous system ODE). Suppose also that x p is one (particular) solution of (Inhomogeneous system ODE). Then the general solution of (Inhomogeneous system ODE) is x = x c + x p , as c varies over all vectors in R n . So we need to find just one solution to (Inhomogeneous system ODE), and then the Superpo- sition Theorem (and what we know about homogeneous systems) finds all the rest. There are two important methods for solving (Inhomogeneous system ODE). The first is un- determined coefficients : to guess the form of a solution, as a formula involving some unknown coefficients; to plug your guess into (Inhomogeneous system ODE); and then to solve the resulting equations for the unknown coefficients. If you can do that, you have found a solution. If you can’t, the conclusion is that your guess about the form of the solution was incorrect. This method in practice is extremely similar to the undetermined coefficient problems you did in solving n th order inhomogeneous equations. Typically there are lots more unknown coefficients (several for each coordinate of your solution guess) so the linear algebra is a bit longer; but there are no essential new ideas. The account in EP 5.8, pages 420–423, is a perfectly reasonable one; so I won’t saynew ideas....
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- Fall '09
- Derivative, inhomogeneous, Inhomogeneous system ODE, Inhomogeneous Superposition Theorem