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y = c (^) 1 e x
y ' ' y = 0 , y ( 0 ) = 0 , y ' ( 0 ) = 1
x t ( ) = c 1 cos t + c 2 sin t x ' ' +
2
x = 0 ( , )
x ( 0 ) = x 0 x ' ( 0 ) = x 1
x t ( ) = x 0 cos t +
x (^) 1
(^) sin t
( x 2 ) y ' ' + 3 y = x , y ( 0 ) = 0 , y ' ( 0 ) = 1
2 < x < < x < 2 2 < x < 2
2 < x < < x < 2
y = c (^) 1 e
x cos x + c 2 e
x sin x ; y ' ' 2 y ' + 2 y = 0
y ( 0 ) = 1, y ' ( ) = 0
y ( 0 ) = 1, y ( ) = 1
y ( 0 ) = 1, y
y ( 0 ) = 0 , y ( ) = 0
( ^ ,^ )
f 1 ( x ) = x , f 2 ( x ) = x
2
, f 3 ( x ) = 4 x 3 x
2
1 The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial value problem.
2 Given that is the general solution of on the interval ,
show that a solution satisfying the initial conditions , is given by
(use a separate sheet to answer if necessary)
3 Find an interval centered about x = 0 for which the given initial value problem has a unique solution.
a. c. e.
b. d.
4 The given two parameter family is a solution of the indicated differential equation on the interval. Determine whether a member of the family can be found that satisfies the boundary conditions.
5 Determine whether the given set of functions is linearly independent or linearly dependent on the interval.
( ,^ )
f 1 ( x ) = 1 + x , f 2 ( x ) = x , f 3 ( x ) = x
2
y (^) 1 = x
3 y (^) 2 = x
3
x
2
y '' 4 xy ' + 6 y = 0 ( , )
5 y ' ' ' + 6 y ' ' y ' + 8 y = 16 , y ( 1 ) = 2 , y ' 1( ) = 0 , y ' ' 1( ) = 0
x ' ' + 16 x = 0
x = c 1 cos 4 t + c 2 sin 4 t
x ' ' + 16 x = 0 , x ( 0 ) = 0 , x
^
f 1 ( x ) = cos
2
x , f 2 ( x ) = sin
2
x , f 3 ( x ) = sec
2
x , f 4 ( x ) = tan
2 x
( 0 ,^ ) f 2
f 1 , f 3 , and f 4
f 1 ( x ) = x + 6 , f 2 ( x ) = x + 6 x , f 3 ( x ) = x 1, f 4 ( x ) = x
2
y 1 = e 3 x y 2 = e 3 x y ' ' 9 y = 0
( ^ ,^ )
6 Determine whether the given set of functions is linearly independent or linearly dependent on the interval.
a. linearly dependent b. linearly independent
7 Are the functions and linearly independent solutions of the differential
equation on the interval.
a. yes b. no
8 Solve the initial value problem
The answer is ________.
9 The two parameter family of solutions of the differential equation is
Solve the boundary value problem
10 Determine whether the given set of functions is linearly dependent or independent on the interval.
11 Show that the given set of functions is linearly dependent on the interval by expressing as a linear combination of
.
12 The functions and are both solutions of the homogeneous linear equation on
the interval. Form the general solution.
y=^1 2
e x
e x
x ( 0 ) = x 0 = c 1
x t ( ) = x 0 cos t + c 2 sin t
x ' ( t) = x 0 sin t + c 2 cos t
x ' ( 0 ) = x 1 = c 2
c 2 =
x (^) 1
x t ( ) = x 0 cos t +
x (^) 1
(^) sin t
no solution dependent f 2 =f 1 +6f 3 c 1 e 3x +c 2 e 3x
4x 2 +5e 2x +x e x
1. 2.
From we see that and . Then implies
. Thus
.
3. e 4.
yes no yes yes
5. dependent 6. b 7. a 8. y= 9. 10. 11. 12.
13.
ANSWER KEY Page 1
ANSWER KEY
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