Exam 5: Modeling With Higher-Order Differential Equations

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

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

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

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

The solution of the differential equation of the previous problem is

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

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

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

The solution of the problem given in the previous problem is

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

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

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

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

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

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

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

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

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

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

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