Question

6. Solve an ODE Using Laplace Transforms: For this problem you are to use Laplace Transforms. Find the complete solution for the initial value problem yº+w2y = t +u.(t - Ttcost, y(0) = 1, y(0) = 0. Hint: Look carefully at the second forcing term and rewrite cost. You can solve this by brute force using the integral below. It would be a good exercise to make sure both approaches give the same Laplace transform. The integral The solution ſeat cosbxdx = Re{elatis)xdx} Relatige(+48)2}

Answer 1

y" + w2y = t + ust-10) cosct) yo 1, Yo)-0 Y"+w'y = t + elect-r cost= t quolt-to) Cosit-T) y'!+w'4 = t- cos (t -71) uct-1) Apply the L í (cosatect's (thult)) = 11 shta? ..Lifct-a)) = e as fos). 5348)-59-9-909 tw? 4+5)= Ś - (5++w2) 1999 - SCI) = 5 SE Sels S. X S2+1 ::70) 84wt), a yes: stis2wt) is 341 354we) + Shwe Sets + S 5²462

1 Apply Inverse LT (Y61) 2-cica) Hoteli sa vening ter s SZEW2 - 1 Lisinat) are (Si sta2 yees welt - Sintot Junts towe [eosce-1) - escorts Xuct-ti) + cosswt) ult) COS(+-11).- Cost y(t) = (Cosewt) +t; - sincwt) jull ws + Śl- cesce ) - cesewet-1)] ust- |gct? = (coswt) to sind ut? Jure) the (cost + cosewet-1))) uct-1) : w211