Question

7. (7 pts) Consider the initial value problem y" +4y' +8y=80), (0=6, yO=0, ſo if 0 <i<6 where g(t) = 8e-21-6) if6 <i<e. (1) Take the Laplace transform of both sides of the given differential equation to create the corresponding alge- braic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). (2) Solve your equation for Y(s). Y(s) = L {y(t)}= (3) Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). Use hat-a) for the Heaviside function shifted a units horizontally. (Class notes have uat)=h(t-a).) yt)= (4) In your written work be sure to show all the steps you took to obtain your answers above. Include your partial fractions decomposition (PFD) and the expression for Y(s) that allows you to easily invert this to y(t). (All elements need to come from our Laplace Table.)

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Answer 1

dion Date : ? Page No.: Date: 3 Applying laplace transfrorm on given dequation, ... $PYCs) – syco) - y.co) +4{scs) – y(0)}+84(s) = Lłg(t)}. 1 a +45 +8} ¥(s) – 65 –242 1 seset est dt (8+45+8) YCS) – 6(5+4)= 82 cup ē (st2it dt = 802 r ecstest you -= 82[{oj - testo] 1.-C8+45+8) Y(s) -6(sk) s eos sta . YCs) = 8ees 6(5+4) (sta)(²+45+8) +45+8 - By taking partial tractions, -istrot -(5+2) Jo 26 1 Yrs) = 6S r 2 - 2(5+2) 7+ 6(5+2) 12 now applying inverse laplace transform, y(t) = 2 12-24. Het-6) - 2 2 -21 Cos (27-12) H (4-6) + 6 e 2 coszt + 6 ezt sinat.