Question

QUESTION 1 (15 MARKS) a) Given 4'{+93}=LC }-( - siu (au) sin’au) sin(2t - 2u) du. Use the convolution theorem to determine the value of constant a. (5 marks) b) Using Laplace transform, solve the simultaneous differential equations dac dt - 4 =1+t, +2=t-1. dt given that x(0) = 0 and y(0) = 3. (10 marks)

Answer 1

on Taking investse Laplace freamfarem. both sides we get ['{x(s)} I'll " + [4 kg is sti b ,الم x(t) = 3 sint & (5 3) * + Again solving (3) and (1) Deget sx(s) – 4() = să 3²4(S) = -5 El Subtracting -($10)Y(s) = -35+1 IXS 5x(s) It Gilt 35 (-) -) کر -) ara 1 y (5) + 3S St1 3²71 5?(5+1 si 5 + 35 1 S²+1 का st 3 S 3²+1 tu on both sides we Taking inverse Laplace transform eget ['74(s)} = 36 3117 Y-575 S S²71 erl, 1(E) 3 cast at 6 ③ and ② are the required solution.

solution (a) [k S $74) 1 S I'Y 5+2? 5²4 22 ) s and g(s) = 5²722 5²72² Coart +(t) s +22 s Let § (s) then I'{I (3)} = I'Y and [X(s)} = ('45*2.5 = sinatzge We Know , by convolution theorcem {"15"{}(3)4 = $(t) and E'{] (s)} = glt) [{F(s).ğ(s) flu) g(t-u) du then z Now, ['{ $123 3 N s²4 2² + ק Conau. Į sin 2(t-u) du Concu sin (21-24) du t ע 2 t 2 t 1-asinu sin (21-24) du [cas 24 = 1-2sinki sin (21-24) du cey5444 - Sinaru) sin (21-24) dy Therefore the value of a is ( z . sin u) 7 Given 1. (Ans)

(b) The is da an 2 3) given differential equation Itt d È - y = d y + x = tai Taking Laplace treanstorem of ① on both sides we get L{ dis-lyg=4419 +L{t} on, sx(s) - X(0) – y(s) = x(0) -y(s) = 3 + an, 52(s) – y(s) = 5 + 5a ["260)20] Taking Laplace treansform of on get Hy alf2yLyt}-44' at, sy(s) - Y60) + x(5) - 52 - Ś at, sy(s) + x(s) a Com [:460)=3 solving (3) and (4), 3²x(s) - Sy(s) = 1 + 3 x(s) +59(s) * 1 -Ś Adding, (5+1)x(3) = 4+32 x(S) = 32 (571) - - 4 both sides we 1 +3 -- S 3x s 40*1 +3 + , اكن 4 S²+1 4 3+1 و اله x(s) = 3