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hmwk3

hmwk3 - solve the following initial value problem...

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Department of Chemical Engineering University of California, Santa Barbara Eng 5A Fall 2001 Instructor: David Pine Homework #3 Homework due: Wednesday, 17 October 2001 Reading: Mathematica Primer Chapter 4 1. (a) Use the DSolve function of Mathematica to ﬁnd the general solutions to the fol- lowing second order linear diﬀerential equations: y 00 ( t ) + y ( t ) = e - t x 00 ( t ) + 4 x ( t ) = 12 t s 00 ( t ) - as 0 ( t ) + 2 s ( t ) = 4 t (b) Find and plot the particular solutions to the above diﬀerential equations for the initial conditions: y (0) = 1 y 0 (0) = 0 x (0) = 2 x 0 (0) = 7 s (0) = 2 + a s 0 (0) = 2 + a Your plots should include the points where the initial conditions are speciﬁed and a large enough range to clearly indicate the basic features of the solutions. For the third plot, you will have to choose a particular value for a ; use a = 2. 2. (a) Using Mathematica as an aid
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Unformatted text preview: , solve the following initial value problem analytically using the method of undetermined coeﬃcients: y 000 ( x ) + 1 2 y 00 ( x ) + y ( x ) = 0 y (0) = y (0) = 0 , y 00 (0) = 1 Notes: You may wish to use numerical expressions for the roots of the characteristic equation just as was done in demo5 in class (and available on the class web site) in order to avoid messy exact expressions. Remember that complex exponentials can be written as sine and cosine functions. Thus, y = e α + iβ ± e α-iβ can be rewritten as 2 e α cos β or 2 e α sin β . (b) Use Mathematica ’s NDSolve to generate a numerical solution to the problem and use this to check your analytical solution (i.e. overlay the plots of both solutions)....
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