Question

1(a) Given that f(x,y) = 6x²y3 – cosx + 5y - siny + 10x, establish the fact that, the order in which the mixed partials derivatives are taken is immaterial. AN [2] = (b) (i)Find the continuous function whose Laplace Transform is 3 F(p) AP [7] P(p2+4) (ii) Determine [-*[0-3]2+26] AP [2] (P-3)2 +16 (c) Find the solution of the IVP; 2y" - y' – 3y = 4x+ex y(0)=-1. y'(0) = -2 AN [9]

Answer 1

sy sing + 100 12 nky t cosu 1.a) ANY) = 68²y3 - cosmas Now for a 2 uy3 + single tio 1243 fyz 1842 42 45 - cosy tyy 3682y + -sing e faya 36x42 b) y FO P(22+4) 3 (02 +4 -- P2) 4.P (P2+4) S + > 3 = F(6) 3 P ip p2.44 So f(t) = 2 2 L70B 2D2r24] 2 cos(at) "} f(t) .) 10 2" } P-332 + 16] 4 Lt (P-332 +42] 4 4 3t sin(het) O

е ) 2002 -- 20/= -2 е) 21-11-24 г 4 7- е* auritary equation is Then auxilan 9 м2. - М -20 и ? 14 (4 5 1 4 24 - 3 - 4 Ч Во 3 le= © 2, + where G, ez are constant. Sek let Up be particular solution. Then Up 4х2 е 22 — D- 3 (х2) =hек 2(+1)- (р) -3 (2) - не 20-14) + 2 - 0 - 1 - 3 Een 2p2+ зр-2 ( 2 ) - 2 е (1 - 22 + 3D " (12) 2. +p?t 3D 1 – 2e1 2 Ч 2ek 2. 2 4. 1 *[ + (аря зруч](**) Г? + 2 + + 3 -2 +) 24 а) - 9 в) 3 * ----- 2" -2° 1xe" - 2-е-4 е-6% е* — 2e7 (+2 - 4 3 So et te е“ - 2e 2 - е- 13 eҡ "13e\

Now 8(0) =-| G+ G = 12 (1) e-2 2- 12- 6- رم - و = 2 -= (0) - - وو 36- 462 = 34 (6) (1) X 2 +(11) 9 8 = و = 58 = 2 85 في - 2 = ه ه 58 orgh Therefore عب/ نے 3 -(*) e ew - 6xe- ( 3e 2- *ے 3 + A)