REVIEW FOR MIDTERM 2
MINGFENG ZHAO
October 24, 2014
1. Topics in Chapter 2
Topic 1. The characteristic equation: The characteristic equation of ay 00 + by 0 + cy = 0 is ar2 + by + c = 0.
Topic 2. Find general solution to ay 00 + by 0 + cy = 0: To find the general solution to a constant coefficient second
order liner differential equation ay 00 + by 0 + cy = 0:
1) Write the characteristic equation of ay 00 + by 0 + cy = 0:
ar2 + br + c = 0.
2) Find the solutions to the characteristic equation ar2 + br + c = 0.
– Compute b2 − 4ac.
– Solve ar2 + br + c = 0:
∗ If ∆ = b2 − 4ac > 0, then r1 , r2 are two different real numbers, and
√
√
−b
−b
b2 − 4ac
b2 − 4ac
+
, and r2 =
−
.
r1 =
2a
2a
2a
2a
∗ If b2 − 4ac = 0, then we have the same real root, and
r1 = r2 = −
b
.
2a
∗ If b2 − 4ac < 0, then r1 and r2 are two different complex numbers, and
√
√
4ac − b2
b
4ac − b2
b
, and r2 = − − i
r1 = − + i
2a
2a
2a
2a
3) Write the general solution to ay 00 + by 0 + cy = 0:
– If b2 − 4ac > 0, then
y = C1 er1 x + C2 er2 x .
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2
MINGFENG ZHAO
– If b2 − 4ac = 0, then
−b
−b
y = C1 e 2a ·x + C2 xe 2a ·x .
– If b2 − 4ac < 0, then
√
y = C1 e
b
− 2a
·x
cos
4ac − b2
x
2a
√
!
b
− 2a
·x
+ C2 e
sin
!
4ac − b2
x .
2a
Topic 3. Use the undermined coefficients to find a particular solution to 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.
Topic 4. The definition of the Wronskian of two functions: The Wronskian of f and g is defined as:
f (x) g(x) = f 0 (x)g(x) − f (x)g 0 (x).
W (f, g) = f 0 (x) g 0 (x) Topic 5. Use the variation of parameters to find a particular solution to ay 00 + by 0 + cy = f (x): 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.
REVIEW FOR MIDTERM 2
3
Then
yp0 (x)
= u1 (x)y10 (x) + u2 (x)y20 (x)
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)
y2 (x)f (x)
dx + y2 (x)
W (y1 , y2 )
Z
y1 (x)f (x)
dx .
W (y1 , y2 )
where
y1 (x) y2 (x)
W (y1 , y2 ) = y10 (x) y20 (x)
= y1 (x)y20 (x) − y10 (x)y2 (x).
Topic 6. The definition of amplitude: The amplitude of A cos(ωt) + B sin(ωt) is:
√
A2 + B 2 .
Topic 7. Mechanical vibrations: The differential equation to the mass-spring system is:
mx00 + cx0 + kx = F0 cos(ωt).
Rewrite the equation, we have
x00 + 2px0 + ω02 x =
F0
cos(ωt),
m
where
c
p=
,
2m
r
and ω0 =
k
.
m
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MINGFENG ZHAO
The characteristic equation of x00 + 2px0 + ω02 x = 0 is:
r2 + 2pr + ω02 = 0.
Solve r2 + 2pr + ω02 = 0, we get
r1,2 = −p ±
q
p2 − ω02 .
Then
• When F0 = 0.
I. Free undamped motion: c = 0 (that is, p = 0)
The general solution to x00 + ω02 x = 0 is:
x(t) = A cos(ω0 t) + B sin(ω0 t) .
II. Overdamping: c2 − 4km > 0 (that is, p2 − ω02 > 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Aer1 t + Ber2 t ,
where
r1 = −p −
q
p2 − ω02 < 0,
and r2 = −p +
q
p2 − ω02 < 0.
III. Critical damping: c2 − 4km = 0 (that is, p2 − ω02 = 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Ae−pt + Bte−pt .
IV. Underdamping: c2 − 4km < 0 (that is, p2 − ω02 < 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Ae−pt cos
q
q
ω02 − p2 t + Be−pt sin
ω02 − p2 t ,
• When F0 6= 0:
I. Undamped (c = 0):
The differential equation for the undamped forced motion (c = 0) is:
x00 + ω02 x =
F0
cos(ωt).
m
REVIEW FOR MIDTERM 2
The general solution to x00 + ω02 x =
x(t) =
5
F0
cos(ω0 t) is:
m



 A cos(ω0 t) + B sin(ω0 t) +
F0
cos(ωt), if ω0 6= ω,
m[ω02 − ω 2 ]


 A cos(ω0 t) + B sin(ω0 t) +
F0
t sin(ω0 t),
2mω0
.
if ω0 = ω
The case of ω0 = ω is called the pure resonance.
II. Dameped (c > 0):
The differential equation for the damped forced motion (c > 0) is:
x00 + 2px0 + ω02 x =
F0
cos(ωt).
m
The transient solution to x00 + 2px0 + ω02 x = 0 is:



Aer1 t + Ber2 t ,
overdamping, that is, p2 − ω02 > 0



Ae−pt + Bte−pt ,
critical damping, that is, p2 − ω02 = 0 .
xtr (t) =

q
q



 Ae−pt cos
ω02 − p2 · t + Be−pt sin
ω02 − p2 · t , underdamping, that is, p2 − ω02 < 0
The steady periodic solution to x00 + 2px0 + ω02 x =
xsp (t) =
F0
cos(ωt) is:
m
2ωpF0
F0 [ω02 − ω 2 ]
· cos(ωt) +
sin(ωt) .
m [(2ωp)2 + (ω02 − ω 2 )2 ]
m [(2ωp)2 + (ω02 − ω 2 )2 ]
The amplitude of xsp is:
C(ω) =
1
F0
·p
.
2
m
(2ωp) + (ω02 − ω 2 )2
Then
∗ If ω02 − 2p2 ≤ 0, then C(ω) has the maximum value at ω = 0. But we assume that ω > 0, so C(ω)
can not attain its maximum, that is,
C(ω) < C(0) =
F0
,
mω02
∀ω > 0.
∗ If ω02 − 2p2 > 0, then C(ω) has the maximum value at ω =
p
ω02 − 2p2 , that is,
q
2F0
F0
1
max C(ω) = C( ω02 − 2p2 ) =
=p
·p
2
2
2
2
m
c [4k 2 − c2 ]
4p [ω0 − p ]
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MINGFENG ZHAO
p
F0
ω02 − 2p2 is called the practical resonance frequency, max C(ω) =
In this case, ω =
·
m
1
p
is called the practical resonance amplitude.
4p2 [ω02 − p2 ]
2. Topics in Chapter 6
Topic 8. Definitions of the Laplace transform and the inverse Laplace transform: Let f (t) be a function on
[0, ∞), then
I. The Laplace transform of f , denoted by L[f ](s), is defined as:
Z ∞
f (t)e−st dt, for all s > 0.
L[f ](s) =
0
II. If F (s) = L[f ](s), the inverse Laplace transform of F , denoted by L−1 [F ](t), is defined as:
L−1 [F ](t) = f (t),
∀for all t > 0.
Topic 9. Definition of the convolution of two functions: Let f (t) and g(t) be two functions on [0, ∞), the
convolution of f and g is defined as:
Z
(f ∗ g)(t) =
t
f (τ )g(t − τ ) dτ.
0
Topic 10. Definition of Heaviside function: Let a be a constant, then

 0, if t < a,
ua (t) =
 1, if t ≥ a.
Topic 11. How to formulate a piecewise function in terms of ua (t)?
In general, let 0 ≤ a1 < a2 < a3 < · · · < an , if


 f1 (t),







f2 (t),





 f3 (t),
f (t) =
 ..

.,






 fn (t),





 fn+1 (t),
if 0 ≤ t < a1 ,
if a1 ≤ t < a2 ,
if a2 ≤ t < a3 ,
..
.,
if an−1 ≤ t < an ,
if t ≥ an ,
then,
f (t) = f1 (t) +
n
X
k=2
fk (t)[uk−1 (t) − uk (t)] + fn+1 (t)un (t).
REVIEW FOR MIDTERM 2
Topic 12. Definition of Dirac delta function: For any continuous function f (t) on (−∞, ∞), we have
Z
∞
δ(t)f (t) dt = f (0).
−∞
Topic 13. How to do the partial fractions?
Topic 14. Six Properties of the Laplace transform:
I. Linearity:
L[af (t) + bg(t)](s) = aL[f (t)](s) + bL[g(t)](s).
That is,
L−1 [aF (s) + bG(s)](t) = aL−1 [F (s)](t) + bL−1 [G(s)](t).
II. First Shifting Property:
L e−at f (t) (s) = L[f (t)](s + a).
That is,
L−1 [F (s + a)] (t) = e−at L−1 [F (s)](t).
III. Transforms of derivatives:
L[f 0 ](s)
= sL[f ](s) − f (0)
L[f 00 ](s)
= s2 L[f ](s) − sf (0) − f 0 (0).
IV. Second Shifting Property:
L[ua (t)f (t − a)] = e−as L[f (t)](s),
for all s > 0.
That is,
L−1 e−as G(s) (t) = ua (t)L−1 [G(s)](t − a).
V. Transform of Integrals:
t
Z
L
0
L[f (t)](s)
f (τ ) dτ (s) =
.
s
That is,
−1
L
Z t
F (s)
=
L−1 [G(s)](τ ) dτ.
s
0
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MINGFENG ZHAO
VI. Transform of Convolution:
L[(f ∗ g)(t)] = L[f (t)] · L[g(t)].
That is,
L−1 [L[f (t)] · L[g(t)]] = f ∗ g(t)
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