Math 286 E1/G1 — Final Exam Practice
This is a mock-up exam for the real one.
1. For the first order linear equation
y 0 + P (x)y = Q(x)
(a) Write down the formula for the integral factor, which is ρ in my lecture note:
R
H=e
P (x)dx
(b) Using the integral factor of above, solve the following equation with initial data
y 0 + 2xy = 2x ,
2
y(0) = 5
2
y = 1 + Ce−x ,
H = ex ,
y = 1 + 4e−x
2
2. For the second order linear equation
y 00 + 7y 0 + 10y = 0
(a) Find two solutions of the differential equation above and use the Wronskian
method to show that they are linearly independent.
y1 = e−5x ,
y2 = e−2x ,
W = 3e−7x 6= 0
(b) Find the particular solution which satisfies the initial data
y(0) = 2 ,
y 0 (0) = −1
y(x) = −e−5x + 3e−2x
3. Using the substitution method to solve the Bernoulli type equation (Section 1.6: 22)
x2 y 0 + 2xy = 5y 4
1/3
7x
−3
v=y , y=
15 + Cx7
4. For the linear differential equation
y 000 + 4y 00 + 4y 0 = 0
(a) Write the characteristic equation and solve
r3 + 4r2 + 4r = 0 ,
r1 = 0, r2 = r3 = −2
(b) Find the general solution of the differential equation
yc = C1 + C2 e−2x + C3 xe−2x
(c) Find a particular solution, where the initial data is y(0) = 2, y 0 (0) = 1, y 00 (0) = −3
y=
9 1 −2x 1 −2x
− e
+ xe
4 4
2
5. Given the non-homogeneous equation
y 00 − 4y 0 − 12y = 2 sin(3x)
(a) Find the solution, yc , to the homogeneous counterpart of this equation
yc = C1 e6x + C2 e−2x
(b) Find a particular solution, yp , to the original equation using the method of variation of parameters.
8
14
yp =
cos(3x) −
sin(3x)
195
195
6. For the un-forced oscillation model
mx00 + cx0 + kx = 0
(a) Find a criteria, with detail, to distinguish the under-damping, critical-damping
and over-damping cases. Write the general solution, yc , for each case.
(b) Write the solution to the inhomogeneous case, when the right hand side is replaced
by F0 cos(ωt) knowing c 6= 0
SEE THE BOOK
7. For the system with parameter λ:
(
y 00 + λy = 0
y(0) = 0 y(−π) = 0
(a) Show that, for negative λ, the system has no non-trivial solution, i.e. there is no
negative eigenvalues.
(b) Show that λ = 144 is an eigenvalue for the system. Find the eigenfunction for
this λ. SEE THE BOOK
8. Determine that the following functions are odd only / even only / both odd and even
/ neither odd nor even .
(a) f (x) = 0 Both
(b) g(x) = sin(x) + cos(−2x) Neither
(c) h(x) = x4 −
1
x2
+ tan(x) x 6=
nπ
2
2
Neither
(d) k(x) = (cos(−2x))3 + (sin(3x)) Even Only
2
(e) m(x) = e−2x Even Only
9. Using the Fourier series technique to solve the second order in-homogeneous linear
differential equation.
y 00 + 2y 0 − 2y = cos2 (x) − sin2 (x) − sin(x)
yp =
x ∈ (0, π)
2
3
3
1
cos(x) +
sin(x) −
cos(2x) +
sin(2x)
13
13
26
13
10. Solve the heat equation by Fourier series.
π
t ∈ [0, ∞) x ∈ [0, ]
2
yt (x, t) = 16 yxx (x, t) ,
π
y(0, t) = y( , t) = 0 , y(x, 0) = 5
2
X 20 1
2
e−64n t sin(2nx)
y=
π n
n odd
11. Solve the wave equation by Fourier series
t ∈ [0, ∞) x ∈ [0, 1]
ytt (x, t) = 4 yxx (x, t) ,
y(0, t) = y(1, t) = 0 ,
y=
y(x, 0) = sin3 (πx) ,
yt (x, 0) = sin(πx)
3
1
1
sin(πx) cos(2πt) − sin(3πx) cos(6πt) +
sin(πx) sin(2πt)
4
4
2π
12. For the matrix


1 1 0




A = 0 2 0


1 0 0
(a) Find A2


1 3 0




A2 = 0 4 0


1 1 0
(b) Find the inverse of A if possible
Does not exist
(c) Find all the three eigenvalues of this matrix and their corresponding eigenvectors.
λ1 = 2 , v1 = (2, 2, 1)T ;
λ2 = 1 , v2 = (1, 0, 1)T ;
λ3 = 0 , v3 = (0, 0, 1)T
13. Solve the in-homogeneous system of equations with initial data


 
 
2t
0
1
e
1
 ~x +   , ~x(0) =  
~x 0 = 
1 0
e−t
0

~x =
2 2t
e
3
1 2t
e
3
+
1 t
e
4
−
1 −t
te
2
+ 41 et + 12 te−t −
14. Find the exponential of the matrix A below,

2 3


0 2

0 0

At
e
2t
e


=0

0
0
2t
e
0
+

2


(b) f (t) = t2 cos(2t)
3 −24s
F (s) = 2s
(s2 +4)3
(c) f (3) (t)
s3 F (s) − s2 f (0) − sf 0 (0) − f 00 (0)
16. Find the inverse Laplace Transform of the followings
Table of Laplace Transform
2t
e
1 3t 6t

 

 
0  0 1 4t  =  0

 
0
e2t
0 0 1
0
(a) f (t) = t2 + e3t − 5 cos(5x)
1
F (s) = s23 + s−3
− s25s
+25
(b) F (s) = s(s22+4)
f (t) = 12 − 21 cos(2t)
7 −t
e
12
i.e. eAt

0


4

2
15. Find the Laplace Transform of the followings
1
(a) F (s) = (s+1)(s−1)
f (t) = 12 e−t − 21 et

1 −t
e
12

2t
3te
2t
e
0
2 2t
6t e



4te 

e2t
2t
f (t)
F (s)
tk eat
k!
(s−a)k+1
sin(at)
a
s2 +a2
cos(at)
s
s2 +a2
tf (t)
−F 0 (s)
f 0 (t)
sF (s) − f (0)
R∞
F (σ)dσ
s
1
f (t)
t
Rt
f (τ )dτ
0
1
F (s)
s
Other Formulas
1. sin(2x) = 2 sin(x) cos(x) cos(2x) = cos2 (x) − sin2 (x)
sin(3x) = 3 sin(x) − 4 sin3 (x) cos(3x) = 4 cos3 (x) − 3 cos(x)
2. sin(α) cos(β) = 21 (sin(α + β) + sin(α − β))
sin(α) sin(β) = 12 (cos(α − β) − cos(α + β))
cos(α) cos(β) = 21 (cos(α + β) + cos(α − β))
3. The Fourier series for the 2L-periodic function f , with value 1 on (0, L), −1 on (−L, 0)
and 0 at 0, L and −L is
jπ
4X1
sin( x)
f (x) =
π j odd j
L
4. The Fourier series for the 2L-periodic function f , with formula x on [0, L) and 2L − x
on [L, 2L) is
iπ
L 4L X 1
cos( x)
f (x) = − 2
2
2
π i odd n
L
5. The Fourier series for the 2L-periodic function f , with formula x on (−L, L) and 0 at
L and −L is
∞
2L X
1
iπ
f (x) =
(−1)j+1 sin( x)
π j=1
n
L