Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 537
Section 15791
HW set 1 a
8/22/2011
(1) Without expanding, prove that the following determinant vanishes:
1 a a 2 bc
1 b b 2 ca
1 c c 2 ab
(2) Prove that:
1 cos A sin A
BC
CA
A B
sin
sin
(a) 1 cos B sin B = 4sin
2
2
2
1 cos C sin C
a+b b+c c+a a b
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 4a
11/19/2014
p 985
2u u
=
+ 2u in the form u = f ( x) g ( y ) . Solve the
x 2 y
u
equation subject to the conditions u = 0 and
= 1 + e 3 y when x = 0 , for all values of y.
x
# 2 Find a solution of the equation
p 999
#11 The ends A and B o
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW 3c solution
10/29/2014
p 932
2 (i) To prove: Pn (1) =(1)n
Proof: Generating function of Legendre polynomials,
1
= z n Pn ( x )
(1 2 xz + z 2 )1/ 2 n =0
1
= setting x =
1,
z n Pn (1) on
(1 + 2 z + z 2 )1/2 n =0
1
= z n Pn ( 1)
1 + z n =0
(
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW set 3d
10/29/2014
p 1028
t
15. If L cfw_ f (t ) = F ( s ) , show that L cfw_ f ( ) = aF (as )
a
16. Show that L cfw_sin kt sinh kt =
2k 2 s
s 4 + 4k 4
p 1034
14. Find L 1cfw_
1 + 2s
( s + 2) 2 ( s 1) 2
20. Find L 1cfw_
1
s a3
23. Find L 1c
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW set 3e
11/05/2014
p 699
9. Expand f ( x) = cos x as a Fourier series in the interval < x <
p 704
4. Find the Fourier expansion of the function defined in one period by the relations
1, 0 < x <
f ( x) =
2, < x < 2
1 1 1
and deduce that 1 +
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 3f
11/10/2014
p 947
#2 Form the partial differential equation by eliminating the arbitrary constants from the
given equation
( x a ) 2 + ( y b) 2 + z 2 =
1
#4
Form the partial differential equation by eliminating the arbitrary constants from
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW 3f
Solutions to select problems
11/17/2014
p 947
#4 Form the partial differential equation by eliminating the arbitrary constants from
the equation z = ax 2 + bxy + cy 2
z = ax 2 + bxy + cy 2
z
z
p
= 2ax + by , q
= bx + 2cy
x
y
(1)
2 z
2 z
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 3g solution
11/19/2014
p 962
#4 To solve: p = eq
Solution: Equation is of the form f ( p, q) = 0
Solution is z = ax + by + c where f ( a, b) = 0
a=
eb
Solution is z = eb x + by + c , b, c arbitrary
#5 To solve: ( x + y )( p + q)2 + ( x y )
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW set 3e
11/12/2014
Solutions to select problems
p 699
9. Expand f ( x) = cos x as a Fourier series in the interval < x <
f ( x) = cos x , which is periodic with fundamental period .
f ( x) = a0 + [a n cos
n =1
n
n
x + bn sin
x]
l
l
f is even
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 3h
Solution to select problems
p 974
2 z
2 z 2 z
#9 2 2
+
=
sin x
x
xy y 2
We first find the complementary solution, the solution to
2 zc
2 zc 2 zc
2
+
=
0,
x 2
xy y 2
( D 2 2 DD + D 2 ) zc = operator notation
0 in
11/19/2014
( D D ) 2 z
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 3g
11/12/2014
p 962
Solve the given equations:
#4 p = eq
#5 ( x + y )( p + q)2 + ( x y )( p q)2 =
1 [Hint: Set x + y X 2 and x y Y 2 ]
=
=
#6
z = px + qy 2 pq
#12 z 2 ( p 2 + q 2 + 1) =
1
#19
p + q =+ y
x
#24
yp + xq + pq =
0
#26
p + q sin x
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 4b
12/01/2014
1. Use the general solution of the wave equation y ( x, t ) = F ( x ct ) + G ( x + ct ) to
solve
c 2 y xx = ytt , 0 < x < , 0 < t <
y ( x,0) = yt ( x,0) = 0, 0 < x <
y (0, t ) = h(t ), 0 < t <
p 992
#3 Solve the boundaryval
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 4c
12/03/2014
p 1005
#4 Solve
= 0 within the rectangle 0 < x < a, 0 < y < b , given that
u xx + u yy
u(0,= u( a,= u( x,= 0 and u( x,= x ( a x )
y)
y)
b)
0)
p1022
#3 The bounding diameter of a semicircular plate of radius a is kept at 0 C an
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
HW 4d
12/08/2014
p 1138
#1 Find the Fourier integral representation of the function
0, x < 0
f ( x) =
= 1 / 2, x 0
e x , x > 0
#8 Solve the integral equation
f ( x ) cos xdx = e
0
R.Tam
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
Ordinary Differential Equations
10/01/2014
Terminology
A differential equation is an equation involving derivatives of the unknown quantity,
the dependent variable(s) as function of the independent variable(s).
If there is only one independent
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700 Applications of Firstorder O.D.E.s
10/01/2014
Applications
Geometric curves can be described by differential equations, the solutions of which
when graphed result in the geometric curves.
Orthogonal trajectories of a given family of plane curv
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
Constantcoefficient linear differential equations
10/06/2014
The general linear differential equation (of any order) takes the form
dmy
d m 1 y
d2y
dy
g(
+ am 1 (t ) m 1 + + a2 (t ) 2 + a1 (t ) + a0 (t ) y =t )
m
dt
dt
dt
dt
We consider first
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
Math 53700
10/08/2014
Power Series solutions to Ordinary Differential Equations
To solve linear ordinary differential equations, we seek a solution of the form of a
power series. By substituting the series into the equation, we solve for the coefficients
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2014
MATH 53700
HW 3b
Solutions to select problems
p 914
d
3. (i) To prove: [ xJ1 ( x)] = xJ 0 ( x)
dx
Proof: From classnotes,
( 1) k
x
( ) 2 k +n
J n ( x) =
k =0 k! ( k + n )! 2
10/22/2014
( 1) k x 2 k +1
J 1 (x ) =
k!(k + 1)!( 2 )
k =0
d
d
[ xJ 1 (x )]
=
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 53700
HW 2a
09/28/2011
p737
Eliminate the arbitrary constants and obtain the differential equations:
#5 y = Ae x + Be x + C
#11 y 2 2ay + x 2 = a 2
#19 Find the differential equation of all circles passing through the origin and having
centers on the
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 537
Section 15791
HW 1b
8/24/2011
(1) Use Cramers Rule to solve the equations:
x + 3y + 6z = 2
(a) 3 x y + 4 z = 9 ,
x 4 y + 2z = 7
2 yz zx + xy = 3xyz
(b) 3 yz + 2 zx + 4 xy = 19 xyz
6 yz + 7 zx xy = 17 xyz
(2) If bl = am n, cm = bn l , an = cl m ,
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 53700
HW 1c
8/29/2011
1 2 2
2
(1) If A = 2 1 2 , compute A 4A 5I ,
2 2 1
2 0 1
(2) If f ( x) = x 5 x + 6, find f ( A), where A = 2 1 3
1 1 0
2
k n
k 1
A=
An =
(3) If
, prove that
0 k
0
R.Tam
nk n 1
kn
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 53700
HW 1d
08/31/2011
(1) Express each of the following matrices as the sum of a symmetric and a
skew symmetric matrix:
4 2 3
(i) A = 1 3 6 ,
5 0 7
a a b
(ii) B = c b b
c a c
4 3 3
(2) Show that the matrix 1 0 1 is its own adjoint.
4 4 3
(3) F
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 53700
HW 1f
09/14/2011
1. Determine whether or not the set of vectors u1 = ( 2 ,1,1,1 ), u 2 = (1, 1, 3, 0),
u 3 = (1, 1, 0, 1 ), u 4 = (2, 1, 1, 1) is orthogonal.
2. (a) Let V denote the space of ntuples of the form (u, 2u, 3u, ., nu). Determine
wh
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 53700
1.
2.
3.
4.
5.
R.Tam
HW 1g
09/19/2011
Find the eigenvalues and corresponding eigenspaces of
1 1 1 1
2 2 2 2
A =
3 3 3 3
4 4 4 4
Determine the eigenvalues and eigenspaces of A10 for
1 1 2
A = 0 1 2
0 0 2
0 1 0
Find the diagonalized form o
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
The U.S. Equal Employment Opportunity Commission
Number
NOTICE
915.002
EEOC
Date
7/27/00
1. SUBJECT: EEOC Enforcement Guidance on DisabilityRelated Inquiries and Medical Examinations of
Employees Under the Americans with Disabilities Act (ADA)
2. PURPOSE
Advanced Engineering Mathematics for Engineers and Scientist1
MATH 537

Fall 2011
MATH 537
8/22 8/24/2016
HW set 1 a
(1) Without completely expanding, prove that the following determinant vanishes:
1 a a 2 bc
1 b b2 ca
1 c
c 2 ab
(2) Prove that:
1 cos A sin A
(a) 1 cos B sin B 4sin
1 cos C sin C
ab
bc
ca
B C
CA
A B
sin
sin
2
2
2
a
b
c