Q: A ball is dropped from a height of $$2$$ meters with gravity $$g = 9.8 m/s$^2. $$What is the velocity of the ball when it hits the ground?

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 2} \approx 6.26 m/s$$

Question 1

For which integer values of $$n is$$ $$y(x) = x^n a $$solution to the differential equation $$x y'' - y' - \frac{3y}{x} = 0$$?

Accepted Answer

$$n = 3$$

Question 2

A ball is thrown straight up with initial velocity $$v_0 = 5$ m/s. Gravity is the only force acting, with g = 9.8$ m/s^2$ downward. Which of the following is the correct first-order ODE and initial condition for the velocity v(t$$)$$?

Accepted Answer

$$\frac{dv}{dt} = -9.8$$, $$v(0) = 5$$

Question 3

Which of the following ODEs is linear?

Accepted Answer

$$\frac{df}{dz} + e^z f = z^2$

Question 4

Suppose a boat moves at $$100 m/s $$and the resistive force is proportional to $$\sqrt{v}. $$After $$2$$ seconds, the speed is $$64 m/s. $$Which ODE correctly models the velocity $$v(t)$$?

Accepted Answer

$$\frac{dv}{dt} = -k \sqrt{v}$$, $$v(0) = 100$$

Question 5

Given the ODE $$y'' + 25y = 0$$, for which values of $$\omega is$$ $$y = \cos(\omega t) a $$solution?

Accepted Answer

$$\omega = 5$$

Question 6

Suppose $$b^2 - 4ac$$ \u003c 0$$ for the quadratic equation ax^2$ + bx + c = 0. $$What is the imaginary part of the roots?

Accepted Answer

$$\frac{\sqrt{4ac - b^2$}}{2a}$$

Question 7

A mass-spring-dashpot system is critically damped. The characteristic equation has a repeated root $$r. $$What is the general solution for the position $$x(t)$$?

Accepted Answer

$$x(t) = (c_1$ + c_2$ t) e$$^{rt}$$

Question 8

Given the ODE $$y'(t) = y^2$ t^2$ with y(0) = 3$, what is the explicit solution?

Accepted Answer

$$y(t) = \frac{3}{1 - t^3$}$$

Question 9

A mass-spring-dashpot system is underdamped with roots $$r_1$$ = \alpha + i\beta$$ and r_2$ = \alpha - i\beta. $$What is the general solution for $$x(t)$$?

Accepted Answer

$$x(t) = e$$^{\alpha t}$$ (c_1$ \cos(\beta t) + c_2$ \sin(\beta t))$$

Question 10

Suppose a mass-spring system has position $$y(t) = 6e$$^{-2t}$$ - 3t e$$^{-2t}$$. At what time does the mass pass through the equilibrium position?

Accepted Answer

$$t = 2$$

Question 11

For the ODE $$y'' + 9y = 0$$ with $$y(0) = -1$$, $$y'(0) = 0$$, what is the maximum distance from equilibrium the mass will reach?

Accepted Answer

$$1$$

Question 12

A mass-spring-dashpot system with $$m = 1$$, $$c = 4$$, $$k = 5$$, $$x(0) = 2$$, $$x'(0) = -2 is $$set in motion. Which of the following best describes the position $$x(t)$$ for $$t \u003e 0$$?

Accepted Answer

$$x(t) = e$$^{-2t}$$(2 \cos(t) - 2 \sin(t))$$

Question 13

Suppose the position of a mass in a spring-mass-dashpot system is modeled by $$x'' + c x' + 9x = 0$$ with $$x(0) = 1$$, $$x'(0) = 0. $$For which values of $$c$$ will the mass oscillate and pass through equilibrium infinitely many times?

Accepted Answer

$$0 \u003c c \u003c 6$$

Question 14

A mass-spring-dashpot system is described by $$x'' + 25x = 17 \cos(2\pi t). If$$ $$m_2 = 1$ kg does not produce resonance, which mass should you choose to demonstrate resonance?

Accepted Answer

$$m_1$$ \u003c 1$$ kg

Question 15

Solve the IVP $$y'' + 9y = u(t-1) + \delta(t-1)$$, $$y(0) = 0$$, $$y'(0) = 0. $$What is the Laplace transform of the right-hand side?

Accepted Answer

$$\frac{e$$^{-s}$$}{s} + e$$^{-s}$$

Question 16

Given the ODE $$y'' + 4y' + 3y = 0$$, for which values of $$r is$$ $$y = e$$^{rt}$$ a solution?

Accepted Answer

$$r = -1 or$$ $$r = -3$$

Question 17

A ball is dropped from a height of $$2$$ meters with gravity $$g = 9.8 m/s$^2. $$What is the velocity of the ball when it hits the ground?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 2} \approx 6.26 m/s$$

Question 18

Suppose $$b^2 - 4ac$$ \u003c 0$$ for ax^2$ + bx + c = 0. $$Why must the two roots be complex conjugates?

Accepted Answer

Because the quadratic formula yields $$x = \frac{-b \pm i\sqrt{4ac - b^2$}}{2a}$$, which are conjugates.

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