WOLFRAM|DEMONSTRATIONS PROJECT

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

​
part
1
2
p
0
q
1
a
2
b
3
form of particular solution
derivatives
substitute derivatives
system of linear equations
solution of the equations
particular solution
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
.