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Unformatted text preview: 18.03 Lecture #26 Nov. 6, 2009: notes Topics for today are (first) Duhamels principle (meant to be on Wednesdays lecture, but not covered then) and (second) an introduction to the Laplace transform. For the first topic, here is an enlarged version of the last theorem introduced on Wednesday. Theorem. Suppose f and y are (possibly generalized) functions vanishing for t < , and p is any polynomial. Then (writing D = d dt as usual) we have p ( D )( f y ) = f ( p ( D ) y ) . ( Differentiating convolutions ) Suppose in particular that y is a solution of a differential equation p ( D ) y = g, with g vanishing for t < . Then f y is a solution of the differential equation p ( D )( f y ) = f g. The proof is (formally) very easy: to differentiate ( f y ) = integraldisplay f ( s ) y ( t s ) ds with respect to t , you just differentiate y under the integral sign; thats the only place that t appears. Corollary (Duhamels principlesee EP 4.6). Suppose w is a (generalized) function satisfying the differential equation p ( D ) w = , w ( t ) = 0 ( t < 0) . Then if f is any (generalized) function vanishing for t < , the solution of the differential equation p ( D ) y = f, y ( t ) = 0 ( t < 0) ( Forced ODE for positive time ) is given by y ( t ) = ( f w )( t ) = integraldisplay t f ( s ) w ( t s ) ds ( t 0) . ( Duhamels formula ) Definition. A weight function for the differential equation (Forced ODE for positive time) is a solution w of p ( D ) w = , w ( t ) = 0 ( t < 0) . (Weight function for p ) Duhamels principle says that calculating weight functions is very important for solving forced differential equations, now with arbitrary input f that begins at time zero: the solution just requires computing the single definite integral in Duhamels formula. Heres a way to find the weight function. (Well see another way to think about weight functions using the Laplace transform.) 1 Theorem (fundamental solution of constant coefficient linear ODE). Suppose p is a polynomial of degree n with leading coefficient 1 . Then the solution w to the ODE (Weight function for p ) may be calculated as follows. For t < , w ( t ) = 0 . For t , w ( t ) is equal to the solution to the homogeneous differential equation p ( D ) y = 0 satisfying y (0) = 0 , y (0) = 0 , ... , y ( n 1) (0) = 1 . ( Delta initial conditions ) The function w has n 2 continuous derivatives; the n 1 st derivative jumps from to 1 at t = 0 . Proof. Let y be the response to the unit step input discussed on Monday: p ( D ) y = u , y ( t ) = 0 ( t < 0) . Differentiating this equation with respect to t gives on the right (by (Derivative of Heaviside step function), so p ( D ) y = , y ( t ) = 0 ( t < 0) ....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.
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