Given the differential equation y" + 4 y' + 7y = 9 e-2 cos(3 x) %3D And the Solutions to its Characteristic Equation Characteristic Solutions: R = -2 + 3 i (1) Determine the General Form of the Particular Solution, and (2) Write the General Solution for the Particular Side of the Differential Equation. y, = [Ax + B]e-2 cos(V3x) + [Cx + D]e-2i sin(/3 x) B yp = A e-2* cos(3 x) + Be-2* sin(V3 x) Yp = A e-2 + Bе-2 D y, = Ax e-2* cos( V3 x) + Bx e-2* sin 3 x) y , = [Ax? + Bx]e-2 cos(V3 x) (3 x) + [Cx2 + Dx]e-24 sin y, = Ax e-2 + Bx e-2x

Given the differential equation y" + 4 y' + 7y = 9 e-2 cos(3 x) %3D And the Solutions to its Characteristic Equation Characteristic Solutions: R = -2 + 3 i (1) Determine the General Form of the Particular Solution, and (2) Write the General Solution for the Particular Side of the Differential Equation. y, = [Ax + B]e-2 cos(V3x) + [Cx + D]e-2i sin(/3 x) B yp = A e-2* cos(3 x) + Be-2* sin(V3 x) Yp = A e-2 + Bе-2 D y, = Ax e-2* cos( V3 x) + Bx e-2* sin 3 x) y , = [Ax? + Bx]e-2 cos(V3 x) (3 x) + [Cx2 + Dx]e-24 sin y, = Ax e-2 + Bx e-2x

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

q17

Given the differential equation
y" + 4 y' + 7y = 9e-2 cos( 3 x)
And the Solutions to its Characteristic Equation
Characteristic Solutions: R = -2 + V3 i
(1) Determine the General Form of the Particular Solution, and (2) Write the General Solution for the Particular Side of the
Differential Equation.
A
= [Ax + B]e- cos(V3x) + [Cx + D]e=2* sin(3 x)
-2 sin(/3 x)
!!
B
y, = A e-2* cos(3x) + Be-2 sin3x)
3 х
O y, = A e-2
+ Be-2r
D y, = Ax e -2* cos( V3 x) + Bx e-2* sin( V3 x)
+ Bx e-2 sin( 3 x)
E
y, = [Ax? + Bx]e-2 cos(3 x) + [Cx? + Dx]e * sin(/3 x)
!!
F
= Ax e-2x
+ Bx e-2

Expert Solution

Step 1

The operator method is very convenient in using the particular solution. Every linear differential equation of higher order can be expressed as $F (D) y = f (x)$ .

Suppose $y_{p}$ be the particular integral.

Then,

$y_{p} = \frac{f (x)}{F (D)}$

There are many rules for different $f (x)$ . For an example, when $f (x) = e^{a x}$ ,

$y_{p} = \frac{e^{a x}}{F (a)}, F (a) \neq 0$