Question

Consider the initial value problem Remaining ume: 111.20 ( mm:sec 0 y' +6y= if 0 <t<1 if 1 <t < 5 if 5 <t<o, 10 0 y(0) = 8. (a) Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y by Y. Do not move any terms from one side of the equation to the other (until you get to part (b) below). (b) Solve your equation for Y. Y =L{y} (C) Take the inverse Laplace transform of both sides of the previous equation to solve for y. y =

Answer 1

y'+6y= 0 10 solution page - ostal iltis to 5 ctcoo It is given differential equation with y colI8 lo Now, Note that ostal letas = funct) - ugct ] 10 set Loo where ų (t) denote unit step function so given differential equation can be written as y'toy - [u, (t)- Ug (t)] 10 =) y'+69 lou, ct)- 10 ug ct) Now Apply laplace transform on both sides above equation we get Ley') +6LCY) = 10 lcuict)) - 10 L Cus(t)) Now, we have Lly) = SY-yo) 2 Luct)) = s s from 0 f @ we get sy-ycol +64 = 10 -55 lo S S Y [s- 67-8 lo S - [ēs ē55]

page 2 Y CS-6) = 10 10 [es- e55] + 8 S c es ess] + 8 Y = lo SCS-6) s-6 =) Y- 10 és S (5-6) 255 SCS-6) + 8 5-6 Now en bothsides Apply laplace inverse = 10 [[y] ess 머 S CS-6) SCS-6) 8. 5-6 yet) = lois es ] 6 S-6 5 -100 2-55 6 [s-6 S + 8 its I s-6 yet) = lo es esto ]} - oi ess[ šo s}} to eat 2

page :( Now TJ Vse I. S est 1 S-6 f Let get) = esti Now By using (fettcs)} = Ųct). 9c+=0) where [ {f(s)} = get) én ☺ eef -]} - 4 Ct).[e "] Ug (t) [e 6(t-1) کم ㅗ S-6 S & cress Isto 6(t-s) s 80 equation ③ becomes 6ct-) yct) = 10 6 [4, ct) [es 1]]-10[44) [ec-?]] + sest Ho'oycta fufct) [erct-123] - uçct) [et_ps] trert It is the solution of given initial value problem