Deck 5: Modeling With Higher-Order Differential Equations

Question 1

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

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

Answer 1

none of the above

Question 2

A rocket is launched vertically upward with a speed $v _ { 0 }$ . Take the upward direction as positive and let the mass be m. Assume 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. Let $y ( t )$ be the distance from the center of the earth at time, $t$ . the correct differential equation for the position of the rocket is

A) $m \frac { d y } { d t } = v _ { 0 } - k / y ^ { 2 }$
B) $m \frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$
C) $\frac { d y } { d t } = v _ { 0 } - k / y ^ { 2 }$
D) $\frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$
E) none of the above

Answer 2

$m \frac { d ^ { 2 } y } { d t ^ { 2 } } = - k / y ^ { 2 }$

Question 3

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

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

Answer 3

$\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ( L ) = 0 , y ^ { \prime } ( L ) = 0$

Question 4

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

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

Question 5

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

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 6

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

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 7

In the previous two problems, how long does it take for the chain to fall completely to the ground?

A) $t = ( L / g ) ^ { 2 }$
B) $t = g / L$
C) $t = L / g$
D) $t = \sqrt { L / g }$
E) $t = \sqrt { g / L }$

Question 8

If the mass in the previous problem is pulled down two centimeters and released, the solution for the position is

A) $x = 0.02 e ^ { - 2 t } + 0.04 t e ^ { - 2 t }$
B) $x = 2 e ^ { - 2 t } + 4 t e ^ { - 2 t }$
C) $x = 0.02 e ^ { 2 t } - 0.04 t e ^ { 2 t }$
D) $x = e ^ { - 2 t } \sin t$
E) $x = 0.02 e ^ { - 2 t } \cos t$

Question 9

In the previous problem, the function $x$ can be written as

A) $x = 5 \sin ( t + \phi ) , \text { where } \tan \phi = - 5 / 12$
B) $x = 12 \sin ( t + \phi ) , \text { where } \tan \phi = - 5 / 12$
C) $x = 12 \sin ( t - \phi ) , \text { where } \tan \phi = 5 / 12$
D) $x = 13 \sin ( t + \phi ) , \text { where } \tan \phi = 5 / 12$
E) $x = 13 \sin ( t + \phi ) , \text { where } \tan \phi = - 5 / 12$

Question 10

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

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 11

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

A) $2 \mathrm { sec }$
B) $2 \sec ^ { - 1 }$
C) $4 \mathrm { sec }$
D) $4 \sec ^ { - 1 }$
E) $16 \mathrm { sec } ^ { - 1 }$

Question 12

The differential equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0$ is a model for an undamped spring-mass system with a nonlinear restoring force. The initial conditions are $x ( 0 ) = 0$ , $x ^ { \prime } ( 0 ) = 1$ . The solution of the linearized system is

A) $x = \left( e ^ { t } + e ^ { - t } \right) / 2$
B) $x = \left( e ^ { t } - e ^ { - t } \right) / 2$
C) $x = \cos t$
D) $x = \sin t$
E) $x = \cos t - \sin t$

Question 13

In the previous problem, the solution for $x$ as a function of $t$ is

A) $x = L - \sqrt { L ^ { 2 } - L g t }$
B) $x = L - \sqrt { L ^ { 2 } + L g t }$
C) $x = L - \sqrt { L ^ { 2 } - L g t ^ { 2 } }$
D) $x = L - \sqrt { L ^ { 2 } + L g t ^ { 2 } }$
E) $x = L - \sqrt { L g t ^ { 2 } - L ^ { 2 } }$

Question 14

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

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 15

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

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 16

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

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 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

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

The solution of a vibrating spring problem is $x = 5 \cos t - 12 \sin t$ . The amplitude is

A) 1 $7$
B) $- 7$
C) 7
D) 13
E) 60

Question 19

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

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 20

The solution of the differential equation of the previous problem is

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

Question 21

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

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 22

The moment of inertia of a cross section of a beam is $I$ , and the Young's modulus is $E$ . Its flexural rigidity is

A) $E / I$
B) $E I$
C) $I / E$
D) $E I _ { k }$ , where $k$ is the curvature
E) $k / E I$ , where $k$ is the curvature

Question 23

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

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 24

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

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

Question 25

The solution of the differential equation of the previous problem is

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 26

A spring attached to the ceiling is stretched one foot by a four pound weight. The value of the Hooke's Law spring constant, $k$ , is

A) 4 pounds per foot
B) 1/4 pound per foot
C) 1/4 foot-pound
D) 4 foot-pounds
E) none of the above

Question 27

The solution of the problem given in the previous problem is

A) $x = 15 - 15 ( 1 - 4 \sqrt { 10 } t / 15 ) ^ { 2 }$
B) $x = 15 / 2 - 15 ( 1 - 4 \sqrt { 10 } t / 15 ) / 2$
C) $x = 15 / 2 - 15 ( 1 - 2 \sqrt { 10 } t / 15 ) ^ { 2 } / 2$
D) $x = 15 / 2 - 15 ( 1 - 4 \sqrt { 10 } t / 15 ) ^ { 2 } / 2$
E) $x = 15 - 15 ( 1 - 4 \sqrt { 10 } t / 15 )$

Question 28

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

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 29

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

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

Question 30

A beam of length $L$ is simply supported at one end and free at the other 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

A) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
B) $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
C) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } E I , y ( 0 ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
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 31

The differential equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x \cos x = 0$ is a model for an undamped spring-mass system with a nonlinear forcing function. The initial conditions are $x ( 0 ) = 0.1$ , $x ^ { \prime } ( 0 ) = - 0.1$ . The solution of the linearized system is

A) $x = 0.1 \cos t$
B) $x = 0.1 \sin t$
C) $x = 0.1 \cos t - 0.1 \sin t$
D) $x = 0.1 e ^ { - t }$
E) $x = 0.1 e ^ { t }$

Question 32

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

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 33

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

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

Question 34

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

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 35

In the previous problem, the function $x$ can be written as

A) $x = 3 \sin ( t + \phi ) , \text { where } \tan \phi = - 4 / 3$
B) $x = 4 \sin ( t + \phi ) , \text { where } \tan \phi = - 4 / 3$
C) $x = 5 \sin ( t + \phi ) , \text { where } \tan \phi = - 4 / 3$
D) $x = 5 \sin ( t + \phi ) , \text { where } \tan \phi = - 3 / 4$
E) $x = 4 \sin ( t + \phi ) , \text { where } \tan \phi = - 3 / 4$

Question 36

In the previous two problems, the correct differential equation for the position, $x ( t )$ , of the mass at a function of time, $t$ ,is

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

Question 37

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

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 38

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

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

Question 39

If the mass in the previous problem is pulled down two feet and released, the solution for the position is

A) $x = 2 \cos ( 4 \sqrt { 2 } t ) + 2 \sin ( 4 \sqrt { 2 } t )$
B) $x = 2 \sin ( 4 \sqrt { 2 } t )$
C) $x = 2 \cos ( 4 \sqrt { 2 } t )$
D) $x = 2 \sin ( 4 t )$
E) $x = 2 \cos ( 4 t )$

Question 40

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) $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } + 32 x = 160$
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) $x \frac { d x } { d t } \frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x ^ { 2 } } { d t } = 160$