Question

1. Solve the following Differential Equations.

a) y" - y’ – 2y = 5e2x b) y" +16 y = 4 cos x c) y" – 4y'+3y=9x² +4, y(0) =6, y'(0)=8

2. Use the variation of parameters method to find the general solution to the given differential equation.

3.

Answer 1

page i 2.2 @ yl'-y !-2y = 5 e 2x auxiliarly ean to solve homo. diff equation yll - 91-2y zo is m2 - m -2=0 m²-2m+ m -2 to mcm-2) + (m-1=0 (mt) (m-2) =0 m = -1 m 2 Yo = cle- sc tee 22 for solaton we Particular operator Twe know pax : Weax FCD) Fra) Fcal =D ocean then lear FID) Fica) he Heae a - 2 Fco) = D 2 - 1 - 2 FC2) = 0 F'CO) = 2D - 1 Fical = 4-2=3 5 e 2x 5c e 220 2 26-D-2 3

(Nagpur Universit: S- ons. Method of Variation of Par cosec page2 = clex teze2x + 5 e 2x (b = 4 cosx yu+t6y auxiliary egn for homogenous ean yut 16 9 = o is diff mz+16=0 m = +41 =cl cos4x + c251048 for cue cese ope&ator method To= cosax cosar F(-02) 1-02 FCD2) Cosx Yp= = 4 cosx - 1+16 D2 +49 16 cosol 15 y Yo typ y = cl cos 4x + casin 4x + 4 cosx 15

раа ез Оy" - 4узу 9х2 4 со саlts) = 8 auxiliarely ean is m2 - 4 m +8 EO rm 2 - 3 m - m + 3 = D rm C -3) - 1 (от -3) = 0 Cm - cm-1) = 0 rm = 3 y c = cle3x tezex now for Чр cue operator melhod 3x 2 +4 2 -чD +2 age 2 +4 1 + C D 2 - Ср? ) 4 1 + Та эх? - 4 (р.а) 1 - 2 - D) 23 24 (18) 4D ) ( 3х2 +4) ion are bitte 3 ј (13) +4 (18x) 3 = 9x 2 + 4 - + 24х Up - 9х2 + 24 - 2

Mersin: Summer 2005) CauchyEuler En C.F. = A cos X + B sinx COS X, pages denency soln is y = yet YP Cle 3x trexo + 9x2 +24x-2 yl: 30183x + czex to 1881 24 cltca - 2 citc2=8 - -ci) Y(0) = 8 = 3C1 +2 +24 30tc2= - 16 cltc2=8 301 to2 - 16 - 201= 24 cl=-12 C2= 20 20ex - 12 e3x + 9x2+248-2

Pages yllty = tanec D (D 2+2) = tan? auxiliary ean is m2+1=0 m = ti Yc- CI CO SOC+02 SIDY y = cosx 42 SIC 41. 421-ylly2 = Coslaots in2 = 1 P. I = uyit vyz x = tan 2x U Yox di ( - Sinx tan2x de S -sind sin2x da Cosx er - I sinbx dx Cosx San Sn3x 3510x dr ca SIDX (3-c0322) dx cosoc cosx = t - sinx dx = dt

COS X, P.L. pade Page No. Data I-22 s! t 1 t -t at toot -t2 2 U = loa cosa cos296 2 V- Yltanza v = 8 cosa sin2x dz cos2x SN24 de Cosx 1 - costa dal Cosc Cosol dx Cosa seca dac cosado loog / seal & tana) Sind

osec dr - sin page 7 Data Y = uyl tvyz Y = cosx Clog CcOS) 2 it sing Cloa (secat tana) - sino) y = yet yo y = cl cosx tcasins + sinx (loo (searthanx SID cost Clog (cosx) + cos²x 13 X'=AX A - 1 2 2 2 choracteristic education is d2 - (d) + (-6)=0 22-3dtad-6=0v d (4-3)+2(d-3) = hand = id= -2

here - 32. cosec x de U= page 8 (Page to de 3 Х A-315 4 2 - 4x +2y=0 421= 29 22 Ey 2 = 1, y = 2 Vi= 1 For d₂ = -2 (A-2I) x = 0 1 2 2 u 13] 0 x+2y=0 = - 24 h V2 = 27 1 two eigenvectors vi and va are linearly independen,

pade Page No Date 6.6 know by theo Suppose A has a pair of real eigenvallies di idz and associated eigenvectors vi then general solution of linear system X'= AX IS cle dit vi trg edava X(t) = 3t cle 17t cz e 2t 2.X(t) = 2 1 1