Question #285961

Which of the following is the appropriate form for a particular solution of the differential equation y (4) + 3y 00 − 4y = 2x + cos2 x − cosh x ? (a) yp = (A + Bx)e x + (C + Dx) cos(2x) + (E + F x) sin(2x) (b) yp = (Ax + B)e −x + (C + Dx) cos(−x) + (E + F x) sin(−x) (c) yp = Ax + Bx2 + (C + Dx) cos x + (E + F x) sin(−x) + Ge2x (d) yp = Ax + Bx2 + (C + Dx) cos(2x) + (E + F x) sin(−2x) + Ge−x (e) yp = A + Bx + Cx sin(−2x) + Dx cos(2x) + Exex + F xe−x


1
Expert's answer
2022-01-10T16:17:20-0500
y(4)+3y4y=2x+cos2xcoshxy^{(4)}+3y''-4y=2x + \cos2 x − \cosh x

Correesponding homogeneous differential equation


y(4)+3y4y=0y^{(4)}+3y''-4y=0

Characteristic (auxiliary) equation


r4+3r24=0r^4+3r^2-4=0

(r21)(r2+4)=0(r^2-1)(r^2+4)=0

r1=1,r2=1,r3=2i,r4=2ir_1=-1, r_2 =1, r_3=-2i, r_4=2i

The general solution of the homogeneous differential equation is


yh=c1ex+c2ex+c3cos2x+c4sin2xy_h=c_1e^{-x}+c_2e^{x}+c_3\cos2x+c_4\sin2x

Then the appropriate form for a particular solution of the given non homogeneous differential equation is

(e)


yp=A+Bx+Cxsin(2x)+Dxcos(2x)y_p=A+Bx+Cx \sin(-2x)+ Dx\cos(2x)

+Exex+Fxex+ Exe^{x} +Fxe^{-x}


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