PRACTICE EXERCISES FOR NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
Undetermined Coefficients:
Let a, b and c be constants, consider the equation:
ay 00 + by 0 + cy = f (x).
Let pn (x) and p̃n (x) be polynomials with degree n, a particular solution yp (x) to ay 00 + by 0 + cy = f (x) can be taken as:
f (x)
yp (x)
pn (x)emx cos(kx) + p̃n (x)emx sin(kx)
xα [qn (x)emx cos(kx) + q̃n (x)emx sin(kx)]
where
• α is one of 0, 1 and 2 (α is the multiplicity of m + ki as the solutions to ar2 + br + c = 0):
– If m + ki is not a root of ar2 + br + c = 0, then α = 0.
– If m + ki is a root of ar2 + br + c = 0 and b2 − 4ac 6= 0, then α = 1.
– If m + ki is a root of ar2 + br + c = 0 and b2 − 4ac = 0, then α = 2.
• qn (x) and q̃n (x) are undetermined polynomials with degree n.
Variation of Parameters:
To find a particular solution to the nonhomogeneous equation y 00 + p(x)y 0 + q(x)y = f (x):
I. Find a fundamental set of solutions y1 (x) and y2 (x) to the homogeneous equation y 00 + p(x)y 0 + q(x)y = 0.
II. Let yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x) be a particular solution to y 00 + p(x)y 0 + q(x)y = f (x)
III. Compute yp0 (x), we get
yp0 (x)
= u01 (x)y1 (x) + u02 (x)y2 (x)
+u1 (x)y10 (x) + u2 (x)y20 (x)
Take
u01 (x)y1 (x) + u2 (x)y2 (x) = 0.
Then
yp0 (x)
=
u1 (x)y10 (x) + u2 (x)y20 (x)
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2
MINGFENG ZHAO
yp00 (x)
= u01 (x)y10 (x) + u1 (x)y100 (x) + u02 (x)y20 (x) + u2 (x)y200 (x).
IV. Plug yp (x), yp0 (x) and yp00 (x) into y 00 + p(x)y 0 + q(x)y = f (x), we get
u01 (x)y10 (x) + u02 (x)y20 (x) = f (x).
V. Solve u01 (x) and u02 (x) from the system:


 u01 (x)y1 (x) + u02 (x)y2 (x) = 0

 u0 (x)y 0 (x) + u0 (x)y 0 (x) = f (x).
1
1
2
2
Then
−y2 (x)f (x)
,
W (y1 , y2 )
and u02 (x) =
−y2 (x)f (x)
dx,
W (y1 , y2 )
and u2 (x) =
u01 (x) =
y1 (x)f (x)
.
W (y1 , y2 )
VI. Solve u1 (x) and u2 (x), then
Z
u1 (x) =
Z
y1 (x)f (x)
dx.
W (y1 , y2 )
VII. Write down the solution:
Z
yp (x) = −y1 (x)
where
y2 (x)f (x)
dx + y2 (x)
W (y1 , y2 )
y1 (x) y2 (x)
W (y1 , y2 ) = y10 (x) y20 (x)
Z
y1 (x)f (x)
dx .
W (y1 , y2 )
= y1 (x)y20 (x) − y10 (x)y2 (x).
PRACTICE EXERCISES FOR NONHOMOGENEOUS EQUATIONS
3
Practice Exercises
Exercise 1. Let α be a real constant. a) Find a fundamental set of solutions to y 00 − 4y 0 + α(4 − α)y = 0. (Hint: You
need to consider two cases which depend on α)
b) Compute the Wronskian of those two solutions. (Hint: You need to consider two cases which depend on α)
Exercise 2. a) Find a fundamental set of solutions to y 00 + 10y 0 + 25y = 0.
b) Compute the Wronskian of those two solutions.
Exercise 3. a) Find a fundamental set of solutions to 2y 00 + 4y 0 + 9y = 0.
b) Compute the Wronskian of those two solutions.
Exercise 4. Find the general solution to 2y 00 + 3y 0 + y = −x + 4x2 .
Exercise 5. Find the general solution to 2y 00 + y 0 + 2y = 3x2 .
Exercise 6. Find the general solution to y 00 − 4y 0 + 4y = x.
Exercise 7. Find the general solution to y 00 − 3y 0 = 2x + 4.
Exercise 8. Find the general solution to 2y 00 = 5x2 + 3.
Exercise 9. Find the general solution to y 00 + 2y 0 + 4y = 3e2x .
Exercise 10. Find the general solution to 2y 00 + 4y 0 − 6y = 7ex .
Exercise 11. Find the general solution to y 00 − 6y 0 + 9y = 5e3x .
Exercise 12. Find the general solution to y 00 + 2y 0 − 5y = 3 sin(2x).
Exercise 13. Find the general solution to 2y 00 + 3y 0 + 10y = sin(4x).
Exercise 14. Find the general solution to y 00 + 9y = 2 sin(3x) + cos(3x).
Exercise 15. Find the general solution to y 00 + 3y = 2x2 + xe−3x + sin(3x).
Exercise 16. Find the general solution to y 00 + 4y = 4 sin(2x) + 3.
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MINGFENG ZHAO
Exercise 17. Solve the following problem:
y 00 + y = f (x),
where
y(0) = 0,
y 0 (0) = 1,

 x,
if 0 ≤ x ≤ π,
f (x) =
 πeπ−x , if x > π.
Exercise 18. Find the general solution to y 00 + 4y = (6x + 7) cos(2x).
Exercise 19. Find the general solution to y 00 + 2y 0 + 5y = 3xe−x cos(2x).
Exercise 20. Find the general solution to y 00 + 4y 0 + 4y = x−2 e−2x .
Exercise 21. Find the general solution to y 00 − 2y 0 + y =
ex
.
1 + x2
Exercise 22. Find the general solution to y 00 + 4y = 3 csc(2x).
Exercise 23. Find the general solution to y 00 + y = tan(x).
Exercise 24. Solve the following problem:
y 00 − 2y + y = f (x),
where
y(0) = 0,
y 0 (0) = 0,

 xex ,
if 0 ≤ x ≤ 1,
f (x) =
πx
 esin( 2 ) , if x > 1.
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca