Answer to Question #285961 in Differential Equations for hector12

Question

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

Answer 1

$$y^{(4)}+3y''-4y=2x + \cos2 x − \cosh x$$

Correesponding homogeneous differential equation

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

Characteristic (auxiliary) equation

$$r^4+3r^2-4=0$$

$$(r^2-1)(r^2+4)=0$$

$$r_1=-1, r_2 =1, r_3=-2i, r_4=2i$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{-x}+c_2e^{x}+c_3\cos2x+c_4\sin2x$$

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

(e)

$$y_p=A+Bx+Cx \sin(-2x)+ Dx\cos(2x)$$

$$+ Exe^{x} +Fxe^{-x}$$