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5.5 ConvolutionandtheLaplaceTrans- form

170 CHAPTER 5. laplace TRANSFORMS10. Solvey +2y +4y=f(t), y(0) = 0, y (0) = 0,wheref(t) is givenin the previous Graph the functionf(t) =t (2t 2)u(t 1) + (2t 4)u(t 2) (2t 6)u(t 3) +..12. Solvey +2y +4y=f(t), y(0) = 0, y (0) = 0,wheref(t) is givenin the previous Considerf(t) =e2tmade into a periodic function f(t) by takingfT(t) whereT= 1.(a) Plot f(t) for 0< t <4.(b) FindL[ f(t)](c)y + 2y + 3y= f(t), y(0) = 0, y (0) = 0,14. Use the differentiation theorem to verify thatL[t u(t a)] =e as1s215. Use appropriate theorems to computeL[tsintetu(t a)] Convolution and the laplace Trans-formWe introduce a new operation between two functions called of Two FunctionsLetf(t) andg(t) be two functions.

5.5. CONVOLUTION AND THE LAPLACE TRANSFORM 175 Convolution and Second Order Linear with Constant Coefficients Consider ay 00 +by 0 +cy = g(t), y (0) = c 1, y 0(0) = c 2. If we have the particular solution to the homogeneous yhomo part (t) that sat- isfied the initial conditions y(0) = c1 and y0(0) = c2 then y(t) = yhomo part (t)+ f ∗g(t) will solve the nonhomogeneous IVP.

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  Form, Linear, Second, Order, Convolutions, Transform, Laplace, Nonhomogeneous, Second order linear, Convolutionandthelaplacetrans form, Convolutionandthelaplacetrans, Convolution and the laplace transform

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Transcription of 5.5 ConvolutionandtheLaplaceTrans- form

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