Question

#2 part a b and c please. please write solutions neatly

2. (27 points) Find the general solution of the associated homogeneous equation for each nonhomogeneous differential equation below. Then determine the form of a particular solution y, of the nonhomogeneous equation. Do not solve for the undetermined coefficients in y, (a) (10 points) y" - 9 - 22 y 3x2 (b) (10 points) y" - 4y' + 29y = 8r sin 3x 3 2. (c)points) Find a homogeneous linear differential equation with constant coefficients whose general solution is ye

Answer 1

2X where Dad y'"- gy' -22y = sue (D? GD 222}y = 50e2u Ausilliary Equation to ? Song 6-22=0 D²gD -22=0 (9-11) (0+ 0 (0-11) (0+2)=0 D=11, D=-2 Vociated Complementary Junction, (solution to Homogeneous part) Ge 1D:11 = -2 -2x +Cze Particulare Solutian Yp= Sxean D-11) (D+2) I where ex I v f(DHA) V is any function of x2 fCD) Hoe yp= 5h e 2h (DJCD+2) -2X s (DH-2)-1) (D +(-2) +2) 2n 5e² (D-3) D 56-24 Ő (11-3) 2x (1-08))**

fer (1+(*) *3)*_)* sa pang ( 3+ (%) + (6 +Cjhe.. 3 1 5-21 13 +0+0 - - 13 (2+1/3 ] e 2n J (2 + 1/3 dne [ -24 2 Yp + لبا TB 13 General solution CF + PI -24 -24 Y = Get the ~5 6 y" - uyl +294 = &a Sinin (0²-40 +29 Jy = bn Sinan du Auxilliary Equation 0 40 +2920 D4) 16-4X29 445100 De 4 & 100 g. Da qt si

Complementary function (solution to Homogeneous part) 24 CF e "(( cos(52) + G Sin (52) Yp = Particular folution = fx finan D²4 -4D+29 TP.J from v in any function of n (ce froJv ] + V foo) Here Up= fre Sinza Sofa fin(3W) + d ( B240+29) (6137) D²4D+29 globuenisyen D²-4D 129

8R D 6034 + -20+ 603n (-3) -५0+29 (-49+91- | . 6n3m + (-2014) 6n3m C-350-43+29) . OM 20-40 : 8 _(20+4D) 6m + (-20+५) 63r 40-1601 (20-4D)- Conn 400-16(-3) 8/ (20 +4 +6-242_inn] - - 78 ५ (G6-034 dohn3x +12 60637) ५५ + (५-25) 673 40 216-30-60 . 5061 31+ 3Goy3m 136 + (4-2098734 २० -1100 0.2m

In finish t3n Cash + (ч-р ).. би 32( 8 -SD) (36 4 и 613x Вx Con3 136 ه 6 + (-20) C&tSD) Gизм 31 Сч#25-) (34 Ким 43(m3 + (ч-25) ( 36 м}я +1 Гози) 3- Cұ25 (11) &f (x Snм +7 Созм + 32-бизм (6 (3» — 48 % З 13 ( +96 62м 32 ( 2 09) 4 | Sanм + 2 Ch? 136 + (-26mm +12 %b3м 22 12-39 Ci Sin 3 5 6 73 +3 37 17 2 Съх Ес біѕt t. 2. 509 24

77 771 Y: 9462.etx tę ne =EGX6 * (@tla) e fayl is repeated soot and anxilliary equation occurs only 02 2 Cre and nie. Charact Auxilliary equation (D-0) (0-7)²=0 Differential equation CD (0-7²) yeo Coco²+49 - 14D)/yo Co3 4490 -14D?) yao yll 10 luy" tua yl=0 then y= Summary Y= I Sinansor Casan f(D) if f(-2²) to I sinan or Cosan fo yed Sinan or losan ffma?) 5

Aus 113 Y. Ce 42 13 ① Y = e(Glossu+Gain 5x) + Salinan +34 60532 761 finc3n) + 3 col3m) 786998 12 289 © y" - 1441 +49 ylco
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