Question

Solve by the Method of Undetermined Coefficients. 1. " - 3y' - 4y = 3e2x (ans. y = C1e4x + cze* - e2x) 2. " - 4y = 4e3x (ans. y = C1 e - 2x + C2 e 2x + 4/5 e3x) 3. 2y" + 3y' + y = x2 + 3 sin x (ans. y = ci e-* + C2 e-x/2 + x2 - 6x + 14 - 3/10 sin x- 9/10 cos x) 4. Y" + y' - 2y = 2x, y(0) = 0, y'(0) = 1 (ans. y = ex-he-2x - x - 2) 5. y" + 2y + 5y = 4e* cos 2x, y(0) = 1, y'(O) = 0 (ans. y = e * cos 2x + 72 e-* sin 2x + x e*sin 2x)

Answer 1

--) M:4,-1. PL = 1 in ean Goven y"- uy = 4034 Guven y" - By'- 4y = 3e?" Auxiliary equation m?-3m-450 =m-4m+m-4 = 0 = (-4) + (-4)=0 >(-4) (+1)=0 Thus, the Complementary function (F= Cieant en Now the particular Intigrad 09-30 +4 en- 22-32-4 Thus the general eolution Y(n) = CF +PI = Ciemn + Czēm - 1&2 equration m?- 4 = 0 => m=2,-2 Auxiliary This Complementary frenchon CF = Gie?" + Czean As we know the ein= of f(a)to ! f(a) 3 28 3 een thus Pl= 3.1 4-64 1 e 24 9

Now the particular integral Pl= I 4 en 4.03h 110 02-4 ean Since I ean. f(0) i if f(a) 40. f(a) en = 32 4 3-4 This Pl= 4 0²-4 Thus the general Robution! y(n)= (t +P1 = Cielna cze? Mules 3 Given. 24" + 3y + y = n² +3&ann. Aisillary equation: 2m +3m +1 = 0 → 2m?+ 2m+m+1=0 2m (m+1)+1(m+i)=0 -> (m+1) (2m +1)=0 -> m=-1, m=-112 thus the Complementary function Gien + Cern. Now the particular integral Pls [m² + 3 linn 2D+3D +1 3 rinn + 202 +30 +1 20+30 +1 ? + enn 1+ (209+307 20'+3D+1

- [o-Cap?+30) + (20++30)*-- ] *? + 3 lenin 20' + 3D +1 Since I = l-n+m²+ (-43)+.. In Now Pl= (n - (199+30) m? + (404+90°+ 100) -m+] + 3 rinn 90'+3D +1 - 22 +6 vi] + 0 +92 +0+0+.. + 3 2D+30 +1 &ann Vilka As we know senan - 1 rinan it f(a)to 100) 3 danien Now einn = 2D + 3D +1 2-1)+D+ 3 Runon = 3-( 30+1) einn (30-1)/30+) 3 (30+1) Rinn 90²

Now denn = Renn -10 302-1 Thus. 3-) - 3 (30+) Co i [3 D semn + Roon] renn = renn 20+50 +1 -3 to [scosnt eon] Thus Pl= n²-6n + 14 3 [slasn + evin 10 Therefore general ed. n²-64 +14 e t Y(*) Cien + C en con*- 9 Coen-3 lo 10 - 3 linn 4) Criven y" +yº-2y = in, y(0) = 0, y (0) = 1 Auxiliary equation! m'+m-2 = 0 3) m+2m-m-2=0 m(+2)-1( m+2)=0 - (m+2)(my)=0 → m -2.1 Theis Conflementary factor CF = C, 22n+Caen.

Now the particular integred! P1 = 2n D²+D-2 +1-(00 I-D²+D Since I = 1+ n²tn3+n4+... - this + PI - (0²403 : Jn 04440-] M + 0 + ne .2 - Thus the general keln y (n)= C, ein & Cent (---) y(0)=0 le C,+1=0 - (+C = 1/2 and Y'(n)= -26,820 + zė lė" - 1 ther yılo=0 -26, +G713.1 →-34, +(6+61-42=0 -) -36, + { 220 »-36, = 1312 =) C =- => 2= 9.1

This 5 Given yu+ 2y' +9y= 4e " Coran. ; 4(0)=1, y'(o)=0 Auxiliary equation m? + 2m +5=0 =) mz -2 254-20 2 = -12 thus the complementary factor CF = ē" [S C, COR 2x + C lin 24 Now Particular integral A en Corrn PI= D+2D +5 As we know 1 e an. v ean IN f(0) Visa function of n. +(0+9) This 1 PI = coern 4 en (D-1)2+2 (0-1) +5 n Corrn D'+1-28 + 20-2+5 1 CORRA - 400 D2 + 4

As we know for Coran = и 뽑 senan D*+4? ht Thus А: 1 \- Хот де де" и Casan 6. - и и1 от и Theis y (n) = (FAPI м) 4 **[, а х + (, залах] è n renan Since У(о)= 1, 1 [с, Ф. 1 + (6] += 1 C = 1 Now (+)- е"- с faniм + 2 (4@s 71] -e" [", аѕя 4 (*an 2] En casen 4 сар + [i 4 е? (- твогч) е* )

thus y'(o)=0 then [0+2C,] + [9 +0]+ 1[+0] + +0=0 => 262 +6,=0 => 2/4=(, = 1 - C2 = 2 This + g(n)= em cas2n the lonan the senin