Answer to Question #114624 in Differential Equations for Mildred

Question

Find general equation of (D^2+2D+5)y=34sinxcos
Using method of undetermined coefficientts and d-operator methods.

Answer 1

Let (D²+2D+5)y = 34 sinx cosx = 17 sin(2x) equation 1

The auxiliary equation is given by

D²+2D+5 = 0

D= -1+2i, -1-2i

The complimentary function of equation1 is e-^x [c₁ cos(2x) + c₂ sin(2x) ] equation 2

D-OPERATOR METHOD

particular integral =  $\frac{17}{D^{2}+2D+5}$ sin2x =

$\frac{17}{2D+1} sin2x$

$(2D−1) \frac{17}{4D^{2}-1}sin2x$

(1-2D) sin2x = -4cos2x + sin2x

so, the general solution of equation is

e^-x [c₁ cos(2x) + c₂ sin(2x)] -4 cos2x + sin2x

UNDETERMINED COEFFICIENT METHOD

Let trial solution of equation1 be z= A cos(2x) + B sin(2x) equation3

Dz= -2A sin(2x) + 2B cos(2x) equation4

D²z = -4A cos(2x) - 4B sin(2x) equation5 substituting equations 3,4 and 5 in equation1

(A+4B) cos(2x) + (B-4A) sin(2x) = 17 sin(2x)

comparing like terms on both sides,

A=-4 , B=1

so, z= -4 cos(2x) + sin(2x)

so, the general solution of equation is

e^-x [c₁ cos(2x) + c₂ sin(2x)] -4 cos2x + sin2x