Question

Use this in Exercises 16-21 to find a particular solution. Then find the general solution and, where indicated, solve the initial value problem and graph the solution. 16. y' + 5y' - 6y = 6e3 17. y' – 4y + 5y = 21 18. C/ Gy" +8y' + 7y = 10e-21, y(0) = -2, y0) = 10 19. C/G Y' – 4y + 4y = et, y(0) = 2, y(0) = 0 20. y' +24' +10y = ex/2 21. y' + 6y + 10y = e-3.

16 and 20 please

Answer 1

X of the ODE is given of the form foddy coax them by thosit technique we can easily find the particular solution an, y = ' coor and Ypas Écaje* if (la) to - e end if $($)(a() with car to 16) In the giren problem, y't5y'-(yz6e3x Up = 22.150-0 best [athers Day J 263 [ By (1) ] ... z6x 1 ex alex 1x (or) . I8 3 Now, to find the general solution we considen y" +54'-6420.. e me as trial solution of eat and substituting the yzema in. it we get m²+5m-6202 m² tom-m-620 zo mcm+6)-1(m+6) 20 = (m+6) (m-120 3) mal,-6 "complementary function a yeaciek the be Where c e are arbitray Constanta Hence, genrend solution of ci) in, y z doty les goeier tąé bx + 2 e 3x , case are 2017 y +2y +10yzés -c .cit (0²+ 20 +10) y ze in the given ecuation content)

tion of (ii) wel Now, to find the Comiden enx as 4+2ytloyzo genend solution of trial solution of the and substitutina and substituting Yzem In cili) we aset, m² + 2m tozo . -2 I 12²-4.1.10 [ roots of ax²+bx+c=0 I are xa-bido yac 7 2 2a = mz-2 Id4-40 mz-2 I 5-36 -2 I 6i 22-123i . [where izda ] . Ih atip are root of the auxilang equation Complementary function is ye ex (e. as Bx + (, SirBx) 2. Here, deze" (cicas 3x + C2 Sin3x) A finally the general solution of Cid is % % % % 2 y z ex (ei cas3x+ C₂ Sin3x) + 4 % . cila are con bitrary constants. na ) where