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List of differential equations problems, final exam 1st part | MATH 307, Exams of Differential Equations

Old Dominion UniversityDifferential Equations
Prof. J. Dorrepaal

Final Exam Material Type: Exam; Professor: Dorrepaal; Class: Ordinary Differential Equations; Subject: MATHEMATICAL SCIENCES; University: Old Dominion University; Term: Fall 2005;

Typology: Exams

Pre 2010

Uploaded on 10/18/2008

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Sample FINAL EXAM Part I
MATH 307
Show all your work! No credit will be given for answers only!
1. Find the two power series solutions of the given linear differential equation about
the ordinary point
0x
:
082
yyxy
; then find the solution of the
initial-value problem with the initial conditions:
0)0(,3)0(
yy
.
Answer: The recurrence relation is
,...3,2,1,
)1)(2(
)4(2
2
k
kk
ck
c
k
k
;
02
4cc
Initial value problem solution is:
...)
3
4
41(3
42
xxy
2. When a mass of 2 kg. is attached to a spring whose constant is 32 N/m, it comes
to rest in the equilibrium position. Starting at
0t
, a force equal to
tetf
t
4cos68)(
2
is applied to the system. Find the equation of motion in the
absence of damping.
Hint:
, look for the particular solution in the form
)4sin4cos(
2
tBtAex
t
p
Initial value problem conditions are
0)0(,0)0(
xx
.
Answer:
tetettx
tt
4sin24cos
2
1
4sin
4
9
4cos
2
1
22
1
pf2

Partial preview of the text

Download List of differential equations problems, final exam 1st part | MATH 307 and more Exams Differential Equations in PDF only on Docsity!

Sample FINAL EXAM Part I

MATH 307

Show all your work! No credit will be given for answers only!

1. Find the two power series solutions of the given linear differential equation about

the ordinary point

x  0

y xy y

; then find the solution of the

initial-value problem with the initial conditions:

y  y

Answer: The recurrence relation is

2

k

k k

k c

c

k

k

2 0

c  4 c

Initial value problem solution is:

2 4

y   xx

2. When a mass of 2 kg. is attached to a spring whose constant is 32 N/m, it comes

to rest in the equilibrium position. Starting at t  0 , a force equal to

f t e t

t

( ) 68 cos 4

 2

is applied to the system. Find the equation of motion in the

absence of damping.

Hint:

x c t c t

c

cos 4 sin 4

1 2

, look for the particular solution in the form

( cos 4 sin 4 )

2

x e A t B t

t

p

Initial value problem conditions are

x ( 0 ) 0 , x ( 0 ) 0

Answer:

x t t e t e t

t t

cos 4 2 sin 4

sin 4

cos 4

 2  2

3. Find the general solution for each of the following differential equations:

(A) homogeneous linear differential equations with constant coefficients

(a)

y y y

(b)

y y y

(c)

y   10 y  25 y  0

(B) non-homogeneous linear differential equations

(a)

x

y y y x e

2

 2  

Hint: Look for a particular solution in the following form

x

p

y Ax Bx Cx e

2 2

After substituting this particular solution into the DE you will find that A=1/12,

B=0 and C=0.

(b)

x

y 3 y  2 y sin e

 



Hint: Use variation of parameters to find the particular solution.

Integrals can be solved by substitution.

(c)

x

y y y e

2

  2  12  9

Answer: 3 6

2

2

2

1

xx x

y ce ce xe

4. Cauchy-Euler Equation

2 4 2

x y 4 xy  6 y  2 xx

 



Hint: When determine the right-hand side function don’t forget first to divide by the

coefficient in front of the highest derivative

2

x

, then the function will be

2

f xx

. Use variation of parameters to find the particular solution.

Answer:

3

1 2

ycxc x ln xx