Particular Solution of a Nonhomogeneous Linear Second-Order Differential Equation with Constant Coefficients

Find a particular solution

of the differential equation:

′′

y

(x)+y(x)2x+3

This Demonstration shows the method of undetermined coefficients for a nonhomogeneous differential equation of the form

y''+py'+q=ax+b

with

p

,

q

,

a

, and

b

constants. If

q≠0

, then the form of the particular solution is

cx+d

. If

q=0

and

p≠0

, the particular solution is of the form

x(cx+d)

. If

q=0

and

p=0

, the particular solution is of the form

2

x

(cx+d)

.

The second part shows the solution of a linear nonhomogeneous second-order differential equation of the form

y''+py'+qy=

ax

e

cos(bx)

. Let

r

be a root of the corresponding characteristic equation. If

r≠a±bi

, the particular solution is of the form

ax

e

(ccos(bx)+dsin(bx))

. If

r=a±bi

and

b≠0

, the form is

x

ax

e

(ccos(bx)+dsin(bx))

. If

r

has multiplicity 2, then

r=a

is a real number and the form of particular solution is

c

2

x

ax

e

.