Question 1

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi / 2 ) = 0$ has the solution

Accepted Answer

A)  $y = \sin ( n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
E)  none of the above 
A)  $y = \sin ( n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
E)  none of the above

Question 2

The solution of the differential equation of the previous problem is

Accepted Answer

A)  $\theta = c e ^ { - \mathrm { g } t / l }$ 
B)  $\theta = c e ^ { - l t / g }$ 
C)  $\theta = c e ^ { g t / l }$ 
D)  $\theta = c _ { 1 } \cos ( g t / l ) + c _ { 2 } \sin ( g t / l )$ 
E)  $\theta = c _ { 1 } \cos ( \sqrt { g / l } t ) + c _ { 2 } \sin ( \sqrt { g / l } t )$ 
A)  $\theta = c e ^ { - \mathrm { g } t / l }$ 
B)  $\theta = c e ^ { - l t / g }$ 
C)  $\theta = c e ^ { g t / l }$ 
D)  $\theta = c _ { 1 } \cos ( g t / l ) + c _ { 2 } \sin ( g t / l )$ 
E)  $\theta = c _ { 1 } \cos ( \sqrt { g / l } t ) + c _ { 2 } \sin ( \sqrt { g / l } t )$

Question 3

In the previous problem, the solution for the velocity, $v$ , is

Accepted Answer

A)  $v = v _ { 0 } - k / y ^ { 2 }$ 
B)  $v = v _ { 0 } + k / y ^ { 2 }$ 
C)  $v = c + 2 k / ( m y )$ 
D)  $v = \sqrt { c + 2 k / ( m y ) }$ 
E)  none of the above 
A)  $v = v _ { 0 } - k / y ^ { 2 }$ 
B)  $v = v _ { 0 } + k / y ^ { 2 }$ 
C)  $v = c + 2 k / ( m y )$ 
D)  $v = \sqrt { c + 2 k / ( m y ) }$ 
E)  none of the above

Question 4

A spring attached to the ceiling is stretched 2.45 meters by a four kilogram mass. The value of the Hooke's Law spring constant, $k$ , is

Accepted Answer

A)  1/4 meter-Newton 
B)  4 meter-Newtons 
C)  1/4 Newton per meter 
D)  16 Newtons per meter 
E)  none of the above 
A)  1/4 meter-Newton 
B)  4 meter-Newtons 
C)  1/4 Newton per meter 
D)  16 Newtons per meter 
E)  none of the above

Question 5

In the previous problem, if the mass is set in motion, the natural frequency, $\omega$ ,is

Accepted Answer

A)  $4 \sqrt { 2 } \mathrm { sec }$ 
B)  $4 \sqrt { 2 } \sec ^ { - 1 }$ 
C)  $32 \mathrm { sec }$ 
D)  $32 \mathrm { sec } ^ { - 1 }$ 
E)  $\sec ^ { - 1 }$ 
A)  $4 \sqrt { 2 } \mathrm { sec }$ 
B)  $4 \sqrt { 2 } \sec ^ { - 1 }$ 
C)  $32 \mathrm { sec }$ 
D)  $32 \mathrm { sec } ^ { - 1 }$ 
E)  $\sec ^ { - 1 }$

Question 6

The initial value problem $( L - x ) \frac { d ^ { 2 } x } { d t ^ { 2 } } - \left( \frac { d x } { d t } \right) ^ { 2 } = L g , x ( 0 ) = 0 , x ^ { \prime } ( 0 ) = 0$ is a model of a chain of length $L$ falling to the ground, where $x ( t )$ represents the length of chain on the ground at time $t$ . The solution for $v = \frac { d x } { d t }$ in terms of $x$ is

Accepted Answer

A)  $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) }$ 
B)  $v = \left( \operatorname { Lg } \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) \right) / ( L - x )$ 
C)  $v = \left( \operatorname { Lg } \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) \right) / ( L - x )$ 
D)  $v = \sqrt { L g \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) } / ( L - x )$ 
E)  $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) } / ( L - x )$ 
A)  $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) }$ 
B)  $v = \left( \operatorname { Lg } \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) \right) / ( L - x )$ 
C)  $v = \left( \operatorname { Lg } \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) \right) / ( L - x )$ 
D)  $v = \sqrt { L g \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) } / ( L - x )$ 
E)  $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) } / ( L - x )$

Question 7

A pendulum of length 16 feet hangs from the ceiling. Let $g = 32$ represent the gravitational acceleration. The correct linearized differential equation for the angle, $\theta$ , that the swinging pendulum makes with the vertical is

Accepted Answer

A)  $\frac { d \theta } { d t } + 2 \theta = 0$ 
B)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + 2 \theta = 0$ 
C)  $\frac { d \theta } { d t } + \theta / 2 = 0$ 
D)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + \theta / 2 = 0$ 
E)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } - 2 \theta = 0$ 
A)  $\frac { d \theta } { d t } + 2 \theta = 0$ 
B)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + 2 \theta = 0$ 
C)  $\frac { d \theta } { d t } + \theta / 2 = 0$ 
D)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + \theta / 2 = 0$ 
E)  $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } - 2 \theta = 0$

Question 8

The boundary value problem $T \frac { d ^ { 2 } y } { d x ^ { 2 } } + \rho \omega ^ { 2 } = 0 , y ( 0 ) = 0 , y ( L ) = 0$ is a model of the shape of a rotating string. Suppose $T$ and $\rho$ are constants. The critical angular rotation speed $\omega = \omega _ { n }$ , for which there exist non-trivial solutions are

Accepted Answer

A)  $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { L } \right) ^ { 2 }$ 
B)  $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { L }$ 
C)  $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { 2 L }$ 
D)  $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { 2 L } \right) ^ { 2 }$ 
E)  $\omega _ { n } = \sqrt { \rho / T } \frac { n \pi } { L }$ 
A)  $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { L } \right) ^ { 2 }$ 
B)  $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { L }$ 
C)  $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { 2 L }$ 
D)  $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { 2 L } \right) ^ { 2 }$ 
E)  $\omega _ { n } = \sqrt { \rho / T } \frac { n \pi } { L }$

Question 9

In the previous problem, if $y ( 0 ) = R$ , the escape velocity is

Accepted Answer

A)  $v _ { 0 } = 2 k / ( m R )$ 
B)  $v _ { 0 } = 2 R / ( m k )$ 
C)  $v _ { 0 } = 2 m / ( k R )$ 
D)  $v _ { 0 } = \sqrt { 2 m / ( k R ) }$ 
E)  none of the above 
A)  $v _ { 0 } = 2 k / ( m R )$ 
B)  $v _ { 0 } = 2 R / ( m k )$ 
C)  $v _ { 0 } = 2 m / ( k R )$ 
D)  $v _ { 0 } = \sqrt { 2 m / ( k R ) }$ 
E)  none of the above

Question 10

The solution of the boundary value problem in the previous problem is

Accepted Answer

A)  $y = \omega _ { 0 } / E I \left( L ^ { 3 } x / 48 - L x ^ { 3 } / 16 + x ^ { 4 } / 24 \right)$ 
B)  $y = \omega _ { 0 } / E I \left( x ^ { 2 } / 2 - L x \right)$ 
C)  $y = \omega _ { 0 } E I \left( L ^ { 3 } x / 48 - L x ^ { 3 } / 16 + x ^ { 4 } / 24 \right)$ 
D)  $y = \omega _ { 0 } E I \left( x ^ { 2 } / 2 - L x \right)$ 
E)  none of the above 
A)  $y = \omega _ { 0 } / E I \left( L ^ { 3 } x / 48 - L x ^ { 3 } / 16 + x ^ { 4 } / 24 \right)$ 
B)  $y = \omega _ { 0 } / E I \left( x ^ { 2 } / 2 - L x \right)$ 
C)  $y = \omega _ { 0 } E I \left( L ^ { 3 } x / 48 - L x ^ { 3 } / 16 + x ^ { 4 } / 24 \right)$ 
D)  $y = \omega _ { 0 } E I \left( x ^ { 2 } / 2 - L x \right)$ 
E)  none of the above

Question 11

The solution of a vibrating spring problem is $x = 4 \cos t - 3 \sin t$ . The amplitude is

Accepted Answer

A)  $7$ 
B)  $1$ 
C)  $25$ 
D)  $5$ 
E)  $- 1$ 
A)  $7$ 
B)  $1$ 
C)  $25$ 
D)  $5$ 
E)  $- 1$

Question 12

A 10 foot chain of weight density 2 pounds per foot is coiled on the ground. One end is pulled upward by a force of 10 pounds. The correct differential equation for the height, $x ( t )$ , of the end of the chain above the ground at time $t$ is

Accepted Answer

A)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
B)  $\frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
C)  $x \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
D)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x } { d t } + 32 x = 160$ 
E)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } = 160$ 
A)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
B)  $\frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
C)  $x \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$ 
D)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x } { d t } + 32 x = 160$ 
E)  $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } = 160$

Question 13

A beam of length $L$ is simply supported at the left end embedded at right end. The weight density is constant, $\omega ( x ) = \omega _ { 0 }$ . Let $y ( x )$ represent the deflection at point $x$ . The correct form of the boundary value problem for this beam is

Accepted Answer

A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime } ( L ) = 0$ 
B)  $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ( L ) = 0 , y ^ { \prime } ( L ) = 0$ 
C)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0$ 
D)  $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$ 
E)  none of the above 
A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime } ( L ) = 0$ 
B)  $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ( L ) = 0 , y ^ { \prime } ( L ) = 0$ 
C)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0$ 
D)  $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$ 
E)  none of the above

Question 14

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0$ has the solution

Accepted Answer

A)  $y = \sin ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
E)  none of the above 
A)  $y = \sin ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$ 
E)  none of the above

Question 15

In the previous problem, the solution for the velocity, $v$ , is

Accepted Answer

A)  $v = - k / y ^ { 2 }$ 
B)  $v = k / y ^ { 2 }$ 
C)  $v = \sqrt { c + 2 k / ( m y ) }$ 
D)  $v = c + 2 k / ( m y )$ 
E)  none of the above 
A)  $v = - k / y ^ { 2 }$ 
B)  $v = k / y ^ { 2 }$ 
C)  $v = \sqrt { c + 2 k / ( m y ) }$ 
D)  $v = c + 2 k / ( m y )$ 
E)  none of the above

Question 16

A rocket with mass $m$ is launched vertically upward from the surface of the earth with a velocity $v _ { 0 }$ . Let $y ( t )$ be the distance of the rocket from the center of the earth at time $t$ . Assuming that the only force acting on the rocket is gravity, which is inversely proportional to the square of the distance from the center of the earth, the correct differential equation for the position of the rocket is

Accepted Answer

A)  $\frac { d y } { d t } = - k / y ^ { 2 }$ 
B)  $\frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$ 
C)  $m \frac { d y } { d t } = - k / y ^ { 2 }$ 
D)  $m \frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$ 
E)  none of the above 
A)  $\frac { d y } { d t } = - k / y ^ { 2 }$ 
B)  $\frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$ 
C)  $m \frac { d y } { d t } = - k / y ^ { 2 }$ 
D)  $m \frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$ 
E)  none of the above

Question 17

In the previous two problems, if the mass is set into motion in a medium that imparts a damping force numerically equal to 16 times the velocity, the correct differential equation for the position, $x ( t )$ , of the mass at a function of time, $t$ , is

Accepted Answer

A)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + x / 4 = 0$ 
B)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 2 x = 0$ 
C)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 4 x = 0$ 
D)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 8 x = 0$ 
E)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 32 x = 0$ 
A)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + x / 4 = 0$ 
B)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 2 x = 0$ 
C)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 4 x = 0$ 
D)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 8 x = 0$ 
E)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 32 x = 0$

Question 18

In the previous problem the corresponding non-trivial solutions for $y$ are

Accepted Answer

A)  $y = \sin \left( \omega _ { n } x \right)$ 
B)  $y = \cos \left( \omega _ { n } x \right)$ 
C)  $y = \sin \left( \omega _ { n } ^ { 2 } x \right)$ 
D)  $y = \cos \left( \omega _ { n } ^ { 2 } x \right)$ 
E)  none of the above 
A)  $y = \sin \left( \omega _ { n } x \right)$ 
B)  $y = \cos \left( \omega _ { n } x \right)$ 
C)  $y = \sin \left( \omega _ { n } ^ { 2 } x \right)$ 
D)  $y = \cos \left( \omega _ { n } ^ { 2 } x \right)$ 
E)  none of the above

Question 19

The boundary value problem $r \frac { d ^ { 2 } u } { d r ^ { 2 } } + 2 \frac { d u } { d r } = 0 , u ( a ) = u _ { 0 } , u ( b ) = u _ { 1 }$ is a model for the temperature distribution between two concentric spheres of radii $a$ and $b$ , with $a < b$ .The solution of this problem is

Accepted Answer

A)  $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$ 
B)  $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$ 
C)  $u = c _ { 2 } - c _ { 1 } / r , \text { where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$ 
D)  $u = c _ { 2 } - c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$ 
E)  none of the above 
A)  $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$ 
B)  $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$ 
C)  $u = c _ { 2 } - c _ { 1 } / r , \text { where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$ 
D)  $u = c _ { 2 } - c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$ 
E)  none of the above

Question 20

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ has the solution

Accepted Answer

A)  $y = \sin ( n \pi x ) , \lambda = ( n \pi ) ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( n \pi x ) , \lambda = ( n \pi ) ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( n \pi x ) , \lambda = n \pi , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( n \pi x ) , \lambda = n \pi , n = 1,2,3 , \ldots$ 
E)  none of the above 
A)  $y = \sin ( n \pi x ) , \lambda = ( n \pi ) ^ { 2 } , n = 1,2,3 , \ldots$ 
B)  $y = \cos ( n \pi x ) , \lambda = ( n \pi ) ^ { 2 } , n = 1,2,3 , \ldots$ 
C)  $y = \sin ( n \pi x ) , \lambda = n \pi , n = 1,2,3 , \ldots$ 
D)  $y = \cos ( n \pi x ) , \lambda = n \pi , n = 1,2,3 , \ldots$ 
E)  none of the above

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi / 2 ) = 0$ has the solution

The solution of the differential equation of the previous problem is

In the previous problem, the solution for the velocity, $v$ , is

A spring attached to the ceiling is stretched 2.45 meters by a four kilogram mass. The value of the Hooke's Law spring constant, $k$ , is

In the previous problem, if the mass is set in motion, the natural frequency, $\omega$ ,is

A pendulum of length 16 feet hangs from the ceiling. Let $g = 32$ represent the gravitational acceleration. The correct linearized differential equation for the angle, $\theta$ , that the swinging pendulum makes with the vertical is

In the previous problem, if $y ( 0 ) = R$ , the escape velocity is

The solution of the boundary value problem in the previous problem is

The solution of a vibrating spring problem is $x = 4 \cos t - 3 \sin t$ . The amplitude is

A 10 foot chain of weight density 2 pounds per foot is coiled on the ground. One end is pulled upward by a force of 10 pounds. The correct differential equation for the height, $x ( t )$ , of the end of the chain above the ground at time $t$ is

A beam of length $L$ is simply supported at the left end embedded at right end. The weight density is constant, $\omega ( x ) = \omega _ { 0 }$ . Let $y ( x )$ represent the deflection at point $x$ . The correct form of the boundary value problem for this beam is

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0$ has the solution

In the previous problem, the solution for the velocity, $v$ , is

In the previous two problems, if the mass is set into motion in a medium that imparts a damping force numerically equal to 16 times the velocity, the correct differential equation for the position, $x ( t )$ , of the mass at a function of time, $t$ , is

In the previous problem the corresponding non-trivial solutions for $y$ are

The boundary value problem $r \frac { d ^ { 2 } u } { d r ^ { 2 } } + 2 \frac { d u } { d r } = 0 , u ( a ) = u _ { 0 } , u ( b ) = u _ { 1 }$ is a model for the temperature distribution between two concentric spheres of radii $a$ and $b$ , with $a < b$ .The solution of this problem is

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ has the solution

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 5: Modeling With Higher-Order Differential Equations

The eigenvalue problem y′′+λy=0,y(0)=0,y(π/2)=0y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi / 2 ) = 0y′′+λy=0,y(0)=0,y(π/2)=0 has the solution

The solution of the differential equation of the previous problem is

In the previous problem, the solution for the velocity, vvv , is

A spring attached to the ceiling is stretched 2.45 meters by a four kilogram mass. The value of the Hooke's Law spring constant, kkk , is

In the previous problem, if the mass is set in motion, the natural frequency, ω\omegaω ,is

A pendulum of length 16 feet hangs from the ceiling. Let g=32g = 32g=32 represent the gravitational acceleration. The correct linearized differential equation for the angle, θ\thetaθ , that the swinging pendulum makes with the vertical is

In the previous problem, if y(0)=Ry ( 0 ) = Ry(0)=R , the escape velocity is

The solution of the boundary value problem in the previous problem is

The solution of a vibrating spring problem is x=4cos⁡t−3sin⁡tx = 4 \cos t - 3 \sin tx=4cost−3sint . The amplitude is

A 10 foot chain of weight density 2 pounds per foot is coiled on the ground. One end is pulled upward by a force of 10 pounds. The correct differential equation for the height, x(t)x ( t )x(t) , of the end of the chain above the ground at time ttt is

A beam of length LLL is simply supported at the left end embedded at right end. The weight density is constant, ω(x)=ω0\omega ( x ) = \omega _ { 0 }ω(x)=ω0​ . Let y(x)y ( x )y(x) represent the deflection at point xxx . The correct form of the boundary value problem for this beam is

The eigenvalue problem y′′+λy=0,y′(0)=0,y′(π/2)=0y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0y′′+λy=0,y′(0)=0,y′(π/2)=0 has the solution

In the previous problem, the solution for the velocity, vvv , is

In the previous two problems, if the mass is set into motion in a medium that imparts a damping force numerically equal to 16 times the velocity, the correct differential equation for the position, x(t)x ( t )x(t) , of the mass at a function of time, ttt , is

In the previous problem the corresponding non-trivial solutions for yyy are

The eigenvalue problem y′′+λy=0,y(0)=0,y(1)=0y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0y′′+λy=0,y(0)=0,y(1)=0 has the solution

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi / 2 ) = 0$ has the solution

In the previous problem, the solution for the velocity, $v$ , is

A spring attached to the ceiling is stretched 2.45 meters by a four kilogram mass. The value of the Hooke's Law spring constant, $k$ , is

In the previous problem, if the mass is set in motion, the natural frequency, $\omega$ ,is

A pendulum of length 16 feet hangs from the ceiling. Let $g = 32$ represent the gravitational acceleration. The correct linearized differential equation for the angle, $\theta$ , that the swinging pendulum makes with the vertical is

In the previous problem, if $y ( 0 ) = R$ , the escape velocity is

The solution of a vibrating spring problem is $x = 4 \cos t - 3 \sin t$ . The amplitude is

A 10 foot chain of weight density 2 pounds per foot is coiled on the ground. One end is pulled upward by a force of 10 pounds. The correct differential equation for the height, $x ( t )$ , of the end of the chain above the ground at time $t$ is

A beam of length $L$ is simply supported at the left end embedded at right end. The weight density is constant, $\omega ( x ) = \omega _ { 0 }$ . Let $y ( x )$ represent the deflection at point $x$ . The correct form of the boundary value problem for this beam is

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0$ has the solution

In the previous problem, the solution for the velocity, $v$ , is

In the previous two problems, if the mass is set into motion in a medium that imparts a damping force numerically equal to 16 times the velocity, the correct differential equation for the position, $x ( t )$ , of the mass at a function of time, $t$ , is

In the previous problem the corresponding non-trivial solutions for $y$ are

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ has the solution