University of Notre Dame
Aerospace and Mechanical Engineering

AME 30314: Differential Equations, Vibrations and Controls I
Fall 2015
Homework 2, due September 9, 2015

Problem 1
Determine the solution to $\dot x = \alpha x$ where $x(0) = -1$ . On the same graph, sketch the solution for $\alpha = -2$ , $\alpha = 0$ , and $\alpha = 2$ .

Problem 2
In dead organic matter, the $C^{14}$ isotope decays at a rate proportional to the amount of it that is present. Furthermore, it takes approximately 5600 years for half of the original amount present to decay.

1. If $x(0)$ denotes the amount present when the organism is alive, determine a differential equation that describes the amount of the $C^{14}$ isotope present if $x(t)$ represents the amount present after time $t$ elapses after the organism dies.
2. In contrast to $C^{14}$ , the $C^{12}$ isotope does not decay and the ratio of $C^{12}$ to $C^{14}$ is constant while an organism is alive. Hence, one should be able to compare the ratio of the two isotopes in a dead specimen to that of a live specimen. Determine how many years have elapsed if the ratio of the amount of $C^{14}$ to $C^{12}$ is $20\%$ of the original value.

Do not look up the formula for half-life and exponential decay problems. The point is to derive the equation in order to relate it to the problem, and then to solve it.

Problem 3
Consider the first-order, linear, variable-coefficient, homogeneous ordinary differential equation

$$\dot x + t x = 0.$$

Does assuming a solution of the form $x(t) = e^{\lambda t}$ where $\lambda$ is a constant work? Why or why not?

Problem 4
Consider the first-order, nonlinear, ordinary differential equation $\dot x + x^2 = 0.$ Does assuming a solution of the form $x(t) = e^{\lambda t}$ work? Why or why not?

Problem 5
As part of a fabrication process, you encounter the following scenario. A vat contains 50 liters of water. In error someone pours 100 grams of a chemical into the vat instead of the correct amount, which is 50 grams. To correct this condition, a stopper is removed from the bottom of the vat allowing 1 liter of the mixture to flow out each minute. At the same time, 1 liter of fresh water per minute is pumped into the vat and the mixture is kept uniform by constant stirring.

1. Show that if $x(t)$ represents the number of grams of chemical in the solution at time $t$ , the equation governing $x$ is
   $$\frac{d x}{d t} = -\frac{x}{50},$$
   where $x(0) = 100$ . How long will it take for the mixture to contain the desired amount of chemical?
2. Determine the equation governing $x(t)$ if the amount of water in the vat is $W$ liters, the rate at which the mixture flows out is $F$ liters/minute (and the same amount of fresh water is added), and the amount of the chemical initially added is $C$ grams.
   

Problem 6
The rate by which people are infected by the zombie plague is proportional to the number of people already infected. Let $x$ denote the number of people infected.¹

1. What is the differential equation describing the number of people infected? Denote the proportionality constant by $k$ . What are the units for $k$ in the differential equation? What is the general solution to this equation? What are the units for $k$ in the solution?
2. If at time $t=0$ there are 100,000 people infected and at time $t=1$ (the next day) there are 225,000 people infected, what is the numerical value of $k$ ?
3. For the value of $k$ determined in the previous part, if at time $t=0$ , one person is infected, how long will it take for the zombie plague to infect every person on earth?
   

Problem 7
Assume that the rate of loss of a volume of a substance, such as dry ice or a moth ball, due to evaporation is proportional to its surface area.

1. If the substance is in the shape of a sphere, determine the differential equation describing the radius of the ball and solve it to find the radius as a function of time.
2. If the substance is in the shape of a cube, determine the differential equation describing the length of an edge of the cube and solve it to find the length of the edge as a function of time.
3. Use the answers from the previous two parts to determine which shape would be better for a given quantity of material if it is desired for it to take as long as possible to evaporate.
   

Problem 8
Match the following differential equations with the solutions plotted in the figure without actually solving the equation. Explain your reasoning for each one.

1. $\dot x + 5 x = 5$ .
2. $\dot x + 4 x = 4$ .
3. $\dot x - 2 x = 2$ .
4. $\dot x + 4 x = -4$ .
5. $\dot x + 2 x = -2$ .

Figure 1: $x(t)$ solutions