2.11: Problems

2.1. Consider the system

\begin{gathered}
x^{\prime}=-4 x-y \\
y^{\prime}=x-2 y
\end{gathered}

a. Determine the second order differential equation satisfied by \(x(t)\).
b. Solve the differential equation for \(x(t)\).
c. Using this solution, find \(y(t)\).
d. Verify your solutions for \(x(t)\) and \(y(t)\).
e. Find a particular solution to the system given the initial conditions \(x(0)= 1\) and \(y(0)=0\)

2.2. Consider the following systems. Determine the families of orbits for each system and sketch several orbits in the phase plane and classify them by their type (stable node, etc.)

a. \begin{aligned}
&x^{\prime}=3 x \\
&y^{\prime}=-2 y
\end{aligned}

b. \begin{aligned}
&x^{\prime}= -y \\
&y^{\prime}=-5x
\end{aligned}

c. \begin{aligned}
&x^{\prime}=2y \\
&y^{\prime}=-3x
\end{aligned}

d. \begin{aligned}
&x^{\prime}=x - y \\
&y^{\prime}=y
\end{aligned}

e. \begin{aligned}
&x^{\prime}=2x + 3y \\
&y^{\prime}=-3x + 2y
\end{aligned}

2.3. Use the transformations relating polar and Cartesian coordinates to prove that

\[\dfrac{d \theta}{d t}=\dfrac{1}{r^{2}}\left[x \dfrac{d y}{d t}-y \dfrac{d x}{d t}\right] \nonumber \]

2.4. In Equation (2.34) the exponential of a matrix was defined.

a. Let

\(A=\left(\begin{array}{ll}
2 & 0 \\
0 & 0
\end{array}\right)\)

Compute \(e^{A}\).

b. Give a definition of \(\cos A\) and compute \(\cos \left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right)\) in simplest form.

c. Prove \(e^{P A P^{-1}}=P e^{A} P^{-1}\).

2.5. Consider the general system

\begin{aligned}
&x^{\prime}=a x+b y \\
&y^{\prime}=c x+d y
\end{aligned}

Can one determine the family of trajectories for the general case? Recall, this means we have to solve the first order equation

\[\dfrac{d y}{d x}=\dfrac{c x+d y}{a x+b y} \nonumber \]

[Actually, this equation is homogeneous of degree 0.] It can be written in the form \(\dfrac{d y}{d x}=F\left(\dfrac{y}{x}\right)\). For such equations, one can make the substitution \(z=\dfrac{y}{x}\), or \(y(x)=x z(x)\), and obtain a separable equation for \(z\).

a. Using the general system, show that \(z=z(x)\) satisfies and equation of the form

\(x \dfrac{d z}{d x}=F(z)-z\)

Identify the function \(F(z)\).

b. Use the equation for \(z(x)\) in part a to find the family of trajectories of the system

\begin{aligned}
x' &= x - y \\
y' &= x + y
\end{aligned}

First determine the appropriate \(F(z)\) and then solve the resulting separable equation as a relation between \(z\) and \(x\). Then write the solution of the original equation in terms of \(x\) and \(y\).

c. Use polar coordinates to describe the family of solutions obtained. You can rewrite the solution in polar coordinates and/or solve the system rewritten in polar coordinates.

2.6. Find the eigenvalue(s) and eigenvector(s) for the following:

a. \(\left(\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right)\)
b. \(\left(\begin{array}{ll}3 & -5 \\ 1 & -1\end{array}\right)\)
c. \(\left(\begin{array}{ll}4 & 1 \\ 0 & 4\end{array}\right)\)
d. \(\left(\begin{array}{ccc}1 & -1 & 4 \\ 3 & 2 & -1 \\ 2 & 1 & -1\end{array}\right)\)

2.7. Consider the following systems. For each system determine the coefficient matrix. When possible, solve the eigenvalue problem for each matrix and use the eigenvalues and eigenfunctions to provide solutions to the given systems. Finally, in the common cases which you investigated in Problem 2.2, make comparisons with your previous answers, such as what type of eigenvalues correspond to stable nodes.

a. \begin{aligned}
&x^{\prime}=3 x-y \\
&y^{\prime}=2 x-2 y
\end{aligned}

b. \begin{aligned}
&x^{\prime}=-y \\
&y^{\prime}=-5 x
\end{aligned}

c. \begin{aligned}
&x^{\prime}=x-y \\
&y^{\prime}=y
\end{aligned}

d. \begin{aligned}
&x^{\prime}=2x+3y \\
&y^{\prime}=-3x+2y
\end{aligned}

e. \begin{aligned}
&x^{\prime}=-4x-y \\
&y^{\prime}=x-2y
\end{aligned}

f. \begin{aligned}
&x^{\prime}=x-y \\
&y^{\prime}=x+y
\end{aligned}

2.8. For each of the following matrices consider the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and

a. Find the fundamental solution matrix.
b. Find the principal solution matrix.

a. \(A=\left(\begin{array}{ll}
1 & 1 \\
4 & 1
\end{array}\right)\)

b. \(A=\left(\begin{array}{ll}
2 & 5 \\
0 & 2
\end{array}\right)\)

c. \(A=\left(\begin{array}{cc}
4 & -13 \\
2 & -6
\end{array}\right)\)

d. \(A=\left(\begin{array}{ccc}
1 & -1 & 4 \\
3 & 2 & -1 \\
2 & 1 & -1
\end{array}\right)\)

2.9. For the following problems

1) Rewrite the problem in matrix form.

2) Find the fundamental matrix solution.

3) Determine the general solution of the nonhomogeneous system.

4) Find the principal matrix solution.

5) Determine the particular solution of the initial value problem.

a. \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=2, \quad y^{\prime}(0)=0\)

b. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\)

2.10. Prove Equation (2.75),

\[\mathbf{x}(t)=\Psi(t) \mathbf{x}_{0}+\Psi(t) \int_{t_{0}}^{t} \Psi^{-1}(s) \mathbf{f}(s) d s \nonumber \]

starting with Equation (2.73).

2.11. Add a third spring connected to mass two in the coupled system shown in Figure 2.19 to a wall on the far right. Assume that the masses are the same and the springs are the same.

a. Model this system with a set of first order differential equations.
b. If the masses are all \(2.0 \mathrm{~kg}\) and the spring constants are all \(10.0 \mathrm{~N} / \mathrm{m}\), then find the general solution for the system.

c. Move mass one to the left (of equilibrium) 10.0 cm and mass two to the right 5.0 cm. Let them go. find the solution and plot it as a function of time. Where is each mass at 5.0 seconds?

2.12. Consider the series circuit in Figure 2.20 with \(L = 1.00 H, R = 1.00 \times 10^2 \Omega, C=1.00 \times 10^{-4}F\), and \(V_0 = 1.00 \times 10^3 V\).

a. Set up the problem as a system of two first order differential equations for the charge and the current.

b. Suppose that no charge is present and no current is flowing at time \(t = 0\) when \(V_0\) is applied. Find the current and the charge on the capacitor as functions of time.

c. Plot your solutions and describe how the system behaves over time.

2.13. You live in a cabin in the mountains and you would like to provide yourself with water from a water tank that is 25 feet above the level of the pipe going into the cabin. [See Figure 2.28.] The tank is filled from an aquifer 125 ft below the surface and being pumped at a maximum rate of 7 gallons per minute. As this flow rate is not sufficient to meet your daily needs, you would like to store water in the tank and have gravity supply the needed pressure. So, you design a cylindrical tank that is 35 ft high and has a 10 ft diameter. The water then flows through pipe at the bottom of the tank. You are interested in the height h of the water at time \(t\). This in turn will allow you to figure the water pressure.

Figure 2.28. A water tank problem in the mountains.

First, the differential equation governing the flow of water from a tank through an orifice is given as

\[\dfrac{dh}{dt} = \dfrac{K - \alpha a \sqrt{2gh}}{A}. \nonumber \]

Here \(K\) is the rate at which water is being pumped into the top of the tank. \(A\) is the cross sectional area of this tank. \(\alpha\) is called the contraction coefficient, which measures the flow through the orifice, which has cross section a. We will assume that \(\alpha = 0.63\) and that the water enters in a 6 in diameter PVC pipe.

a. Assuming that the water tank is initially full, find the minimum flow rate in the system during the first two hours.

b. What is the minimum water pressure during the first two hours? Namely, what is the gauge pressure at the house? Note that \(\Delta P=\rho g H\), where \(\rho\) is the water density and \(H\) is the total height of the fluid (tank plus vertical pipe). Note that \(\rho g=0.434\) psi (pounds per square inch).

c. How long will it take for the tank to drain to \(10 \mathrm{ft}\) above the base of the tank?

Other information you may need is 1 gallon \(=231\) in \({ }^{2}\) and \(g=32.2 \mathrm{ft} / \mathrm{s}^{2}\).

2.14. Initially a 200 gallon tank is filled with pure water. At time \(t=0\) a salt concentration with 3 pounds of salt per gallon is added to the container at the rate of 4 gallons per minute, and the well-stirred mixture is drained from the container at the same rate.

a. Find the number of pounds of salt in the container as a function of time.
b. How many minutes does it take for the concentration to reach 2 pounds per gallon?
c. What does the concentration in the container approach for large values of time? Does this agree with your intuition?
d. Assuming that the tank holds much more than 200 gallons, and everything is the same except that the mixture is drained at 3 gallons per minute, what would the answers to parts a and become?

2.15. You make two gallons of chili for a party. The recipe calls for two teaspoons of hot sauce per gallon, but you had accidentally put in two tablespoons per gallon. You decide to feed your guests the chili anyway. Assume that the guests take 1 cup/min of chili and you replace what was taken with beans and tomatoes without any hot sauce. \([1\) gal \(=16\) cups and \(1 \mathrm{~Tb}=3\) tsp.]

a. Write down the differential equation and initial condition for the amount of hot sauce as a function of time in this mixture-type problem.
b. Solve this initial value problem.
c. How long will it take to get the chili back to the recipe's suggested concentration?

2.16. Consider the chemical reaction leading to the system in (2.111). Let the rate constants be \(k_{1}=0.20 \mathrm{~ms}^{-1}, k_{2}=0.05 \mathrm{~ms}^{-1}\), and \(k_{3}=0.10 \mathrm{~ms}^{-1}\). What do the eigenvalues of the coefficient matrix say about the behavior of the system? Find the solution of the system assuming \([A](0)=A_{0}=1.0\) \(\mu \mathrm{mol},[B](0)=0\), and \([C](0)=0\). Plot the solutions for \(t=0.0\) to \(50.0 \mathrm{~ms}\) and describe what is happening over this time.

2.17. Consider the epidemic model learning to the system in (2.112). Choose the constants as \(a = 2.0 \text{ days}^{-1}\), \(d = 3.0 \text{ days}^{-1}\), and \(r = 1.0 \text{ days}^{-1}\). What are the eigenvalues of the coefficient matrix? Find the solution of the system assuming an initial population of 1,000 and one infected individual. Plot the solutions for \(t = 0.0\) to \(5.0\) days and describe what is happening over this time. Is this model realistic?