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MIT18_03S10_ls4 - LS.4 Decoupling Systems 1 Changing...

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± ± ± ± ± ± LS.4 Decoupling Systems 1. Changing variables. A common way of handling mathematical models of scientific or engineering problems is to look for a change of coordinates or a change of variables which simplifies the problem. We handled some types of first-order ODE’s the Bernouilli equation and the homogeneous equation, for instance by making a change of dependent variable which converted them into equations we already knew how to solve. Another example would be the use of polar or spherical coordinates when a problem has a center of symmetry. An example from physics is the description of the acceleration of a particle moving in the plane: to get insight into the acceleration vector, a new coordinate system is introduced whose basis vectors are t and n (the unit tangent and normal to the motion), with the result that F = m a becomes simpler to handle. We are going to do something like that here. Starting with a homogeneous linear system with constant coefficients, we want to make a linear change of coordinates which simplifies the system. We will work with n = 2, though what we say will be true for n > 2 also. How would a simple system look? The simplest system is one with a diagonal matrix: written first in matrix form and then in equation form, it is u 1 0 u u = 1 u (1) = , or . v 0 2 v v = 2 v As you can see, if the coefficient matrix has only diagonal entries, the resulting “system” really consists of a set of first-order ODE’s, side-by-side as it were, each involving only its own variable. Such a system is said to be decoupled since the variables do not interact with each other; each variable can be solved for independently, without knowing anything about the others. Thus, solving the system on the right of (1) gives 1 t ± ± u = c 1 e 1 1 t (2) , or u = c 1 e + c 2 0 e 2 t . 2 t 0 1 v = c 2 e So we start with a 2 × 2 homogeneous system with constant coefficients, (3) x = A x , and we want to introduce new dependent variables u and v , related to x and y by a linear change of coordinates, i.e., one of the form (we write it three ways): u a b x u = ax + by (4) u = D x , = c d y , .
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