MATH 204
Example 6:
Undetermined Coefficients
Dr. TeBeest
Solve the nonhomogeneous ODE
y 00 − y 0 − 6y = 18x2 − 52 cos 2x .
(N)
NOTE: The input function is g(x) = 18x2 − 52 cos 2x.
STEP 1:
Solve the associated homogeneous problem:
y 00 − y 0 − 6y = 0 .
(H)
We did this in Example 1, so the complementary solution of (N) is
yc = c1 e3x + c2 e−2x .
(C)
STEP 2:
Construct a particular solution yp of (N) by Undetermined Coefficients.
Input Function:
Terms
g = 18x2 − 52 cos 2x
g 0 = 36x + 104 sin 2x
g 00 = 36 + 208 cos 2x
x2 , cos 2x
x, sin 2x
a, cos 2x
List:
a, x, x2 , cos 2x, sin 2x
Q: Do any terms in the List already appear in yc ?
A: No, so we need not modify any terms in the List.
So we seek a particular solution of (N) that is a linear combination of terms in the List:
yp = a + bx + cx2 + d cos 2x + e sin 2x .
(P)
We will substitute yp into (N), but first
yp = a + bx + cx2 + d cos 2x + e sin 2x ,
yp0 = b + 2cx − 2d sin 2x + 2e cos 2x ,
yp00 = 2c − 4d cos 2x − 4e sin 2x .
c 2000-14
Prof. K.G. TeBeest Kettering University
1
file: ex6.tex
05/06/2014
Plug these into (N) to obtain
yp00 − yp0 − 6 yp ≡ 18x2 − 52 cos 2x ,
2c − 4d cos 2x − 4e sin 2x
− ( b + 2cx − 2d sin 2x + 2e cos 2x )
− 6 ( a + bx + cx2 + d cos 2x + e sin 2x )
≡
18 x2 − 52 cos 2x .
Collect like terms:
(−4d − 2e − 6d) cos 2x + (−4e + 2d − 6e) sin 2x
− 6c x2 + (−2c − 6b) x + (2c − b − 6a)
≡
18 x2 − 52 cos 2x .
simplify:
(−10d − 2e) cos 2x + (2d − 10e) sin 2x
− 6c x2 + (−2c − 6b) x + (2c − b − 6a)
≡
0 + 0 x + 18 x2 − 52 cos 2x + 0 sin 2x .
Equate like terms:
x2
x
k
cos 2x
sin 2x
:
:
:
:
:
−6c ≡ 18
−2c − 6b ≡ 0
2c − b − 6a ≡ 0
−10d − 2e ≡ −52
2d − 10e ≡ 0
The latter two equations give
d=5
and
=⇒
=⇒
=⇒
c = −3
b = (−1/3) c = 1
a = (2c − b)/6 = −7/6
e = 1.
Litmus Test: Note that these terms are exactly those terms that were in the “List”.
So by (P), a particular solution of (N) is
yp = a + bx + cx2 + d cos 2x + e sin 2x
7
= − + x − 3x2 + 5 cos 2x + sin 2x .
6
STEP 3:
Then the general solution of the nonhomogeneous problem (N) is
y = yc + yp
= c1 e3x + c2 e−2x −
7
+ x − 3x2 + 5 cos 2x + sin 2x .
6
It is a 2–parameter family of solutions of (N).
STEP 4:
Apply initial conditions to the general solution found in Step 3, NOT to solution (C).
c 2000-14
Prof. K.G. TeBeest Kettering University
2
file: ex6.tex
05/06/2014