Vivian Duong
French0400 Intermediate French
Composition 2
Un de mes amis, John, ressemble Antoine. Cependant, John est un amricain dont
cheveux sont blonds et friss. Il est un tudiant indpendant maintenant avec une petite
amie extraordinaire.
Toutefois, q

Claim: If y = (t) is a solution of the non-homogenous 2nd order dierential
equation
y + p(t)y + q (t)y = g (t)
(1)
where g (t) is not always zero, then
y = c(t)
(2)
is not a solution.
Proof. Because (2) is a solution of (1), then
(t) + p(t) (t) + q (t)(t)

I Show that Lf (ct) =
1s
F( )
cc
Denition 1 (Laplace Transform). F (s) = Lf (t), s > a 0 s > ca
Lf (ct) =
st
e f (ct)dt
0
u
u = ct, = t
c
du
=c
dt
1
du = dt
c
1
Lf (ct) =
c
e
0
su
c f (u)du
t=u
dt = du
Lf (ct) =
1
1s
F( )
cc

Show that
2
ex dx =
0
2
(1)
Let
2
I = 0 ex dx
2
2
I2 = 0 ex dx[ 0 ey dy ]
2
2
I2 = 0 ex ey dxdy
r=
2
x2 + y 2
2
I2 = 0 0 ex y dxdy
2
I2 = 0 02 er rddr
r2
I2 =
e rdr
20
u = r 2
du = 2rdr
du
= rdr
2
u
e du
4 0
r2
[e |0 ]
I2 =
4
2
[limr er 1]
I2 =
4
I2 =

Proof.
dy
Theorem 0.1. Given M (x, y ) + N (x, y )
= 0, My = Nx , and (x, y (x) =
dx
which is the implicit solution.
M dy =
N dx, (x, y ) = c,
x = M
y = N
dy
x + y
=0
dx
x dx + y dy = 0
dx dy
+
=0
x dx
y dx
d
(x, y (x) = 0
dx
Denition 1 (Partial Derivat

The gamma function is denoted by (p) and is dened by the integral
ex xp dx
(p + 1) =
(1)
o
The integral converges as x for all p. For p < 0 it is also improper because the integrand becomes
unbounded as x 0. However, the integral can be shown to converge

Theorem 0.1. Euler equations, equations in the form
t2 y + ty + y = 0
(1)
where t > 0, and are real constants, can be transformed into an equation with constant coecients by
transforming x into lnt.
Proof.
x = ln(t)
(2)
2
t y + ty + y = 0
dy
d2 y
+ y = 0

2nd ODE Repeated Roots
Vivian Duong
November 21, 2012
Prove that the 2nd root of a characteristic equation of repeated roots is ter2 t . Suppose that r1 and r2
are roots of ar2 + br + c = 0 and that r1 = r2 ; then er1 t and er2 t are solutions of the dier

Complex Roots of the Characteristic Equation
Vivian Duong
Prove that y1 = cost and y2 = sint are a fundamental set of solutions of y + y = 0 and that the
Wronskian isnt zero.
Denition 1 (characteristic equation). The characteristic equation of a 2nd order

1st ODE Integrating Factor
Vivian Duong
November 19, 2012
Claim: Suppose the First Order Dierential Equation
dy
+ p(t)y = g
dt
(1)
where p(t) and g (t) are continuous functions. Continuous functions have no
holes or breaks in it. If you were to draw it, y

PHYS0030 HW4
Vivian Duong
November 18, 2012
6.40 A 3.50 x 102 -kg box is pulled 7.00 m up a 30 inclined plane by an external force of 5.00 x 103 N
that acts parallel to the frictionless plane. Calculate the work done by (a)the external force, (b)gravity,

2 Proofs that the Derivative of x to the nth power equals nx to the
n-1 power
Vivian Duong
Two proofs that
dn
x = nxn1
dx
Proof. Here is the rst one.
Let y = xn
ln(y ) = ln(xn )
ln(y ) = nln(x)
d
d
[ln(y (x)] =
[nln(x)]
dx
dx
1
dx dy
.
=n
dy dx
x
1 dy
1
.