Section 3.5. Undetermined Coefficients aka ”Guessing”
Find a particular solution of
y 00 + ay 0 + by = f (x)
NOTE: This method can be used
only when:
1.
constant coefficients
2.
f is an ”exponential” function
1
• If f (x) = cerx
set
z(x) = Aerx
Example:
y 00 − 5y 0 + 6y = 7e−4x.
Set
z = Ae−4x
where A is to be
determined.
2
−4x .
Answer: z = 1
e
6
The general solution of the differential
equation is:
y = C1
e2x
+ C2
e3x +
1 −4x
e
.
6
3
• If f (x) = c cos βx,
or
set
d sin βx,
c cos βx + d sin βx,
z(x) = A cos βx + B sin βx
where A, B are to be determined
4
Example:
y 00 − 2y 0 + y = 3 cos 2x.
Set
z = A cos 2x + B sin 2x
5
9 cos 2x − 12 sin 2x.
Answer: z = − 25
25
The general solution of the differential
equation is:
y=
C1ex+C2xex−
12
9
cos 2x−
sin 2x.
25
25
6
Example:
Find a particular solution
of
y 00 − 2y 0 + 5y = 2 cos 3x − 4 sin 3x + e2x
Set
z = A cos 3x + B sin 3x + Ce2x
where A, B, C are to be determined.
Answer
8
1
1 2x
z=−
cos 3x +
sin 3x + e .
13
13
5
7
• If f (x) = ceαx cos βx,
or
set
deαx sin βx
ceαx cos βx + deαx sin βx
z(x) = Aeαx cos βx + Beαx sin βx
where A, B are to be determined.
8
Example:
y 00 + 9y = 4ex sin 2x.
Set z = Aex cos 2x + Bex sin 2x
4 ex cos 2x+ 6 ex sin 2x.
Answer: z = − 13
13
9
A BIG Difficulty:
The trial solution
z is a solution of the reduced equation.
10
Examples:
1. Find a particular solution z1 of
y 00 + y 0 − 6y = 3e2x.
See Example 3, Section 3.4
11
2. Find a particular solution z2 of
y 00 − 6y 0 + 9y = 5e3x.
12
3. Find a particular solution z3 of
y 00 + 4y = 2 cos 2x.
13
4. Find a particular solution z4 of
y 00 − 5y 0 + 6y = 4e2x + 3
14
5. Find a particular solution z5 of
y 00 − 3y 0 = 4e3x − 2
15
Answers:
3 2x
z1 = x e
5
5 2 3x
z2 = x e
2
1
z3 = x sin 2x
2
z4 =
−4xe2x +
1
2
4 3x 2
z5 = e − x
3
3
16
The Method of Undetermined Coefficients
A.
Applies only to equations of the
form
y 00 + ay 0 + by = f (x)
where a, b are constants and f is an
“exponential” function.
c.f Variation of Parameters
17
B. Basic Case:
• f (x) = aerx
If:
set z = Aerx.
• f (x) = c cos βx, d sin βx, or
c cos βx + d sin βx,
set z = A cos βx + B sin βx.
• f (x) = ceαx cos βx,
deαx sin βx or
ceαx cos βx + deαx sin βx,
set z = Aeαx cos βx + Beαx sin βx.
18
BUT:
•
If
z
satisfies the reduced
equation, try
xz;
• if xz also satisfies the reduced
equation, then x2z
will give a
particular solution.
19
C. General Case:
• If
f (x) = p(x)erx
where p is a polynomial of degree n,
then
set
z = P (x)erx
where P is a polynomial of degree n
with undetermined coefficients.
20
Example:
Find a particular solution
of
y 00 − y 0 − 6y = (2x2 − 1)e2x.
Set z = (Ax2 + Bx + C)e2x.
Answer: z =
2
−1
x
2
−
3x − 9
4
16
e2x.
21
• If
f (x) = p(x) cos βx + q(x) sin βx
where p, q are polynomials of degree
n, then
set
z = P (x) cos βx + Q(x) sin βx
where P, Q are polynomials of degree
n with undetermined coefficients.
22
Example:
y 00 − 2y 0 − 3y = 3 cos x + (x − 2) sin x.
Set z = (Ax+B) cos x+(Cx+D) sin x.
Answer:
47
1
2
1
x−
cos x − x −
sin x.
z=
10
50
5
25
!
!
23
• If
f (x) = p(x)eαx cos βx + q(x)eαx sin βx
where p, q are polynomials of degree
n, then
set
z = P (x)eαx cos βx + Q(x)eαx sin βx
where P, Q are polynomials of degree
n with undetermined coefficients.
24
Example:
y 00 + 4y = 2x ex cos x.
Set
z = (Ax + B)ex cos x + (Cx + D)ex sin x
Answer:
1
1
x
z=
(10x−7)e cos x+ (5x−1)ex sin x.
25
25
25
BUT:
Warning!!!
• If any part of z satisfies the
reduced equation, try
xz;
• if any part of xz also satisfies
the reduced equation, then x2z
will give a particular solution.
26
Examples:
1.
Give the form of a particular
solution of
y 00 − 4y 0 − 5y = 2 cos 3x − 5e5x + 4.
Answer:
z = A cos 3x + B sin 3x + Cxe5x + D
27
2.
Give the form of a particular
solution of
y 00 + 8y 0 + 16y = 2x − 1 + 7e−4x.
Answer:
z = Ax + B + Cx2e−4x
28
3.
Give the form of a particular
solution of
y 00 + y = 4 sin x − cos 2x + 2e2x.
Answer:
z = Ax cos x+Bx sin x+C cos 2x+D sin 2x
+Ee2x
29
4.
Give the form of the general
solution of
y 00 + 9y = −4 cos 2x + 3 sin 2x
Answer:
z = A cos 2x + B sin 2x
30
5.
Give the form of the general
solution of
y 00 + 9y = −4 cos 3x + 3 sin 2x
z = Ax cos 3x+Bx sin 3x+D cos 2x+E sin 2x
31
6.
Give the form of the general
solution of
y 00 + 4y 0 + 4y = 4xe−2x + 3
Answer:
z = (Ax3 + Bx2 )e−2x + C
32
7.
Give the form of the general
solution of
y 00 + 4y 0 + 4y = 4e−2x sin 2x + 3x
Answer:
z = Ae−2x cos 2x+Be−2x sin 2x+Cx+D
33
8.
Give the form of the general
solution of
y 00 + 4y 0 = 4 sin 2x + 3
Answer: z = A cos 2x + B sin 2x + Cx
34
9.
Give the form of the general
solution of
y 00 + 2y 0 + 10y = 2e3x sin x + 4e3x
Answer:
z = Ae3x cos x + Be3x sin x + Ce4x
35
10.
Give the form of the general
solution of
y 00 + 2y 0 + 10y = 2e−x sin 3x + 2e−x
Answer:
z = Axe−x cos 3x+Bxe−x sin 3x+Ce−x
36
11.
Give the form of a particular
solution of
y 00 −2y 0 −8y = 2 cos 3x−(3x+1)e−2x −4
Answer:
z = A cos 3x+B sin 3x+(Cx2 +Dx)e−2x +E
37
12.
Give the form of a particular
solution of
y 00 − 2y 0 − 8y = 2 cos 3x − 3xe−2x − 3x
Answer:
z = A cos 3x+B sin 3x+(Cx2 +Dx)e−2x
+Ex + F
38
13.
Find the general solution of
2x
e
y 00 − 4y 0 + 4y = −4e2x +
x
Answer:
y = C1e2x + C2xe2x − 2x2e2x + xe2x ln x
39