Question

5. Consider the initial value problem ?′′+9?′+18?=3? ,?(0)=2,?′(0)=−1
a. Determine an appropriate ?? for the differential equation.
b. Using concepts from “Undetermined Coefficients,” determine the appropriate ??.
c. Using concepts from “Variation of Parameters,” determine the appropriate ??.
d. Using the results from parts a-c, determine the solution to the initial value problem

Please use the specified method, no matrices

Answer 1

☆ Given y"+9y'+ 18y = 3t 9(0): 2, y'(o)=-1 09+90 +18) y = 30 Auxiliary ean is flm) 20 m² tam+ 18 20 m + 6 m+ 3m +18=0 m(m +6) +3(m +6)=0 m+6)(m+3) =0 m-3 6 ma felx) = C, et test Complementary function is particular integrala Ypx): flo) 3+ D+90 +18 t voline D? +90 (-e'esp) t - Ć (1-601799) +(stole...) (t-0999)+ -0) 5(-104) 1(-+)

undertermined Coefficients CO? + +40 +18) y - 3+ 9p= Az + B&te yp = 2AX+B 9. -ZA 2A +9 (24*+B) +18 (4&+B4+c) = 36 Compare t Coefft Compare Constants 18A +18B 3 2A + 9B +18 CO A+B= 3 18 compare ² Coefft A+B= 16 A=0 +18C-0 B- from © 260) +(*)** hec = -2 2 C=-1 12 from ® Yp B (t) + ((t) - 1 / 3 t 6 등을 general solution is y= 9ct Yp -64 3t y=ce +ge + -

initial Condetion : 9(0)=2, y'(OJ = -1 = t -6+ -3t y=cé t qe alt 6 12 Ylo) y=2 when t=0, - 2 co ) - + Çeo) 2 the -ام a tę - the 2 = atq=25 V 12 2+ iz Cit(₂= worot 7 deff -34 -O -6+ эс, е + 6 1027 y's y'= -60, e y (0) = -1 → when (0)+7 -13-66,8 -3 66 +34 = 1 69, +36=75 -1= -663€¢ 29 +2=

25 12 C+C2 = 띔 L 2C, +% 13 C = 잃 t. 이 36 25 ㅋ 1 -C- 뜷 12 CS SL 61 36 - - 5 alt 25 (2) 156 정 2 3 Ca SL-nl 9. 19 ~ 36 16t -3t e + 134 - 7 + t. 6 - y = 12 e 36 १६