Question

(4) Use any method to solve the equation. (a) y + 5y + 6y = 1862 (b) y + y = sint+t cost (s) Solve the initial value problem. y" + y = sect y(0) = 0,7/(0) = 1

Answer 1

The given equation is , (a) y "+ sy' +by = 134 The auxiliary :eq. is, y " + sy't by = 0 Do +50 + 6 =o (Put g' =D . Now Da + 2D + 3D +6=0 D(D+ 2 + 3 (D 2) = 0 D+2) = 0 D + 3) = 0 D = -2, D= -3 the complementary solution is, y ct) = ce at the at we find the particular solution, Jb = I .CP.4.5.) - f (D) I 18+ Do +50 +6 Now .661 + $492; 18+ = +(1+ ble 9 " [Now we use (lejt = 5(!-(0232) + (P1525'+ -) 188 = 1-xtedt - = 3 [18+?-(36 +5,364) + eg, oud) |[ 0 = Derivative = +[18+"- (6 490+) + 25,36+ --] bc ve t'j = D(35+) - [164 - 6 - 30t +257 D3 (18t² = o.] = 36

= So mil Phange B by -&*] { [ 187" - 30€ +19] complete solution is, y(t) = Cient + 56347 4[184"-33t+19 ] (b) The given eq. is, y"+y - Sint + { cost auxiliary eq. is, - its Del +1 = , DN.-1 D = ti Roots are complex of the form a tiß so complementary solution is of the form. y(t) : evt. co, cos Bt + cq Sinßt) (azo, g(t) = C, cost + s sint Nowe the particular integral is, B = 1) YB - o ß I CR.H.S.) FOD (Sint + t Cost) 1 Da +1 ot, Sint + I tast - 2+1

I, - sint I DH change Dal by-a?] -1 +1 I Sinat, Gjet] A I Sint close of failure) Differentiate the auiliary eq, E do Sint | Cosť [ } = Integrating 12. ct cost factu)= [t-folie =[- Br.cort [t to, cost = [tb] 1. Cost f sint to Sint - 252, (sigt) = t Sint - 8! (-Cost) (DCsint= -loge .tsint + I cost = tsint + Sint DP+1 .