6 15 pts Determine a suitable form for Y if the method of un

6. [15 pts. Determine a suitable form for Y() if the method of unde- termined coefficients is to be used. Do not evaluate the constants. (sin 2t t) -1 y t cost 3e-2t

Solution

ex 6)

given that y⁴+4y\'\"+4y\"=e^-2t(sin2t+t)-1

consider r⁴+4r³+4r²=0

r²(r²+4r+4)=0

i.e r²=0 or r²+4r+4=0

i.e r=0 or (r+2)²=0

r=0 or r+2=0

r=0 or r=-2

therfore genral soution is y_c=C₁e^-2t+C₂

since y⁴+4y\'\"+4y\"=e^-2t(sin2t+t)-1

=e^-2tsin2t +e^-2tt-1

for this y_p=e^-2t(Acos2t+bsin2t)+Ce^-2t+D

therfore sutabel form of Y is

Y(t) =C₁e^-2t+C₂+e^-2t(Acos2t+bsin2t)+Ce^-2t+D

= C₁e^-2t+e^-2t(Acos2t+bsin2t)+Ce^-2t+D₁   where D₁=C₁+D

B) given that

y⁴-3y\"-4y=t²cost -e^-2t

consider auxillary equation

m²-3m-4=0

(m-4)(m+1)=0

i.e m-4=0 or m+1=0

i.e m=4 or m=-1

therfore genral solution is

y_c=C₁e^-t+C₂e^4t

to find perticular solution with constat

give y⁴-3y\"-4y=t²cost -e^-2t

^let y=(At²+Bt+C)(Dcost +Esint )+3Fe^-2t

=(ADt²+BDt+CD)cost +(AEt²+BEt+CE)sint +3Fe^-2t

=(At²+Bt+C)cost +(Dt²+Et+F)sint +3Ge^-2t   (where A,B,C,D,E,F,G are new costant)

therfore solution of given equation is in the form

y(t)=C₁e^-t+C₂e^4t+(At²+Bt+C)cost +(Dt²+Et+F)sint +3Ge^-2t