Question

18-27 IVPs, SOME WITH DISCONTINUoUS INPUT Using the Laplace transform and showing the details, solve 18. 4y"-12y' + 9y-0, y(0)-2/3, y,(0) | 20. у', + IOv, + 24y 14412, y(0) 19/12. y (0)5 th Ze 22. y" +3y' + 2-4t İf 0 < t < 1 and 8 if t > 1; y(0) = 0, y'(0) = 0 23. y" + y,-2y-3 sin t-cos t, (0 < t < 2π), and 3 3 sin 2t - cos 2t, (t2T); y(0) 0, y'(0)1 24. y" + 3y' 2y 1 if 0<1 and 0 if t> 1; 3 y(0) 0, y (0)0 25. y" +y 2t if 0 <I< 1 and 2 if t>1 26. Shifted data. y" + 2y 10sin tif ltTm

problem 20

Answer 1

"Given Ivp is y""+ 10 y' + 24y = 144t^2, g(0) = 19/2, y'(0) = -5 Apply laplace transform on both side of 2 we get L [y"" + 1oy' + 24y] = L[144t^2] L[y""] + 10 L[y'] + 24 L[y] = 144 L [t^2] s^2Y(s) - sy(0) - y'(0) + 10[sY(s) - y(0)] + 24 Y(s) = 144 21/S^3 where Y(s) = L[y(t)] [therefore L (t^n) = n!/s^n + 1) s^2 Y(s) - s middot 19/2 - -(-5) + 10 sY(s) - `10 middot 19/12 + 24 Y(s) = 288/s^3 S^2 Y(s) - 19s/12 + 5 + 10 s Y(s) - 95/6 + 24 Y(s) = 288/s^3 S^2Y(s) + 10 SY(s) + 24 Y(s) - 19/12 S + 30 -95/6 = 288/s^3 (s^2 + 10 s + 24) Y(s) = 288/s^3 + 19/12 s + 65/6 Rightarrow Y(s) = 1/s^2 + 10 s + 24 [288/s^3 + 19/12 s + 65/6] \r\n Rightarrow Y(s) = 1 times (3456 + 19s^4 + 130 s^3)/(s^2 + 10 s + 24) middot 12 s^3 "