Question

a) i. Express in terms of the unit step function, the piecewise continuous causal functions (2t2, Ost<3 F(t) = {t + 4, 3 st<5 9, t25 [3 marks] ii. Use Laplace transforms to solve the initial value problem a) 7" + 16y = 4cos3t + s(t – 1/3) where y(0) = 0 and y'(0) = 0. [7 Marks) E.K. Donkoh (Ph.D) or [7 marks) B) y' – 3y = F(t), where y(0) = 0 and (sint, Osts F(t) = 1, t2 b) i. Form partial differential equation from z = ax - #2y + b [4 marks] ii. Solve the partial differential equation მx2 მ; 18xy2 + sin(2x - y) = 0 c) i. Solve the Lagrange equation [4 Marks)

Answer 1

(i) we've Fit). 2+2 ; 03743 4+4 iz st25 ; 47,5 9 or flt) = 2t?+ U(4-3)(t+4-at? + Ult-5) (94+9)) flt) = 2ť?+ 4ct-3) (t+9-2+2) + 417-5) (9-7-4) (ii) we've (a) yl+169 = 46053t+8[t=73) - O with 4 (0)=0 9 y/o)=0 :ify!") - S" Y 1S)-shdyco)-s= {/10) & l{cosats - 82792 L{817-18)3 = et73) : from Taking laplace on both sid s? yls)-5400)=9'0) +16415)= _45 5432 9 S He is

احاد + $? +9 + e we : ||$41833 = {5779 5416 > s?+16) yls) = 45 Yls), 45 es (8?+9)(82+16) 18+16) Now taking inverse taplace on both side get ... 45 .4517 4517 (5P+9952416) - 5449 82416 4817 4517 e --+43) (82716 - cos 3t - 4c054t+ 4511143-4 -04-2++) ylt) - 4160537 -CO54 +)+ + sin(97-4t) o(t-13) () b) we've nyl-3y=flt) ; 9101=00 where Flt)= s sint lost<9/2 i tena Flt) - Sint + U17-12)(-Sint) taking laplace of F(t) we get 7 f(5) -25/2 Ish cost? or S²+1 + etabys & 3-Sib(b*)) its 2 te
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