Question

In Exercises 1-6 find a particular solution by the method used in Example 5.3.2. Then find the general solution and, where indicated, solve the initial value problem and graph the solution. 1. y' + 5y - 6y= 22 + 180 - 1842 2. y' - 4y + 5y = 1+ 5.0 3. y' + 8y + 7y = -8-2+24x2 + 7ar3 4. y' - 4y + 4y = 2 + 8x - 4.2 CIG /' +2y +10y = 4 +262 +6x2 +10x, y(0) = 2, 10) = 9 C/G/' +64 +10y = 22 + 202, y(0) = 2, 10) = -2 5. 6.

evens from 2 and 6

Answer 1

Soln 2 are have, y"-ay't5y =1+52 Homogeneous ego is y"-by'+5y The charactoristic polynomial of the compleme pemertory epping 82_48+5= r=4185-4% 4x1x5 4 Ir-4 2 2x are fa 2z roof of cherrerforistie eg? - The complementary solubon is yo (21) = (c, cos (et casin (21) The pecticular solution is yp"=0 e Yp=A+bre Yp'z B egn becomes, o-4B+58AA821) = 1+521 => 6413+5A) + 5B 2 = 1+52 coefficient we regel -GB +5A = equelting 5B=5 A- ula => B =), :-Yp=43 + ze » The solution 15 general y = y + y2 =) y =€?*(4,206() + (zsony) + + Re.

© ylltoylaloy = 22 froze ' yio) -2, y'(o) = -2 Homogeneous egno yll try toy,=0 The charactoristic polynomia of the complem ewory egnis r2+ir+10 IV.52_4X1X10 -otra 7c rten 22 2 To regn -37 are roots of chron cachoristic o the complementory solution is -32 Y, 12) = C. cos (21) + sin (21) (2) The pechiular soluson - At Bu Ypilso e 15: Yp = yp's B tegn becomes = 22 +2021 o+6B+ locAABU) => 66 Btion) + 10B21222720 20 weges coethicient on both side equating JO 3 ro F OBTIOA El2 -2 A= 1 16 13 syp= y +221 - The general solution y = 4, typ > y= é 320 del cocoscou) + €zšín (n)) + 1 fese As you jl= c, C- a sincas* 3. x) + celé e cosaa)-3e sinu) -321 -32 -32 -32 co5 + 2 Asyio)=2, ylcov = -2 2 = C, tl f -2= -80 +(2+2 C1) 4025-1 -32, => Y (21) =, e "(costa sinca) +1 +23. which is required solution