# Math 308-302 — Summer 2014 (Narcowich/Battle)

## Homework

Assignment 1

• Problems. Many answers are in the back of the book. Thus, in order to receive credit, you must show all necessary steps in solving the problem. Also, the integrals must be done by hand - no symbolics. Use Matlab to do any required plots.
1. Section 1.1: 3, 7, 10, 23
2. Section 1.2: 9, 13, 15
3. Section 1.3: 5, 6, 13

Due Friday, June 6, 2014.

Assignment 2

• Problems. Many answers are in the back of the book. Thus, in order to receive credit, you must show all necessary steps in solving the problem. Also, the integrals must be done by hand - no symbolics. Use Matlab or your favorite program to do any required plots. Include labels for the axes, the equation plotted, the problem number and your name.
1. Section 2.1: (You may use the exact solution to find the large t behavior.) 6(b,c), 15, 16
2. Section 2.2: 6, 8, 21
3. Section 2.3: 3, 5, 18, 20

Due Wednesday, June 11, 2014.

Assignment 3

• Problems. Many answers are in the back of the book. Thus, in order to receive credit, you must show all necessary steps in solving the problem. Also, the integrals must be done by hand - no symbolics. Use Matlab or your favorite program to do any required plots. Include labels for the axes, the equation plotted, the problem number and your name.
1. Section 2.4: 3, 6, 13, 21
2. Section 2.5: Consider the situation in problem 18. Let $V_0 =\frac{\pi}{3} a^2 h$, which is the total volume of the conical pond. This probelem is aimed at scaling out units.
1. Derive the equation required in 18(a). Show that $\frac{3a}{\pi h}=a^3/V_0$, so the equation becomes $dV/dt = k - \pi \alpha a^2(V/V_0)^{2/3}$. What are the units of $k$, $\alpha$?
2. Let $W=V/V_0$ and $\tau = \beta t$. Find $\beta$ and $A$ for which the equation has the form $dW/d\tau = A - W^{2/3}$. What are the units of $W$, $\tau$, and $A$?
3. Find the equilibrium value $W$. Is this value stable or unstable?
4. Find a condition on $A$ that guarantees the pond will not overflow. Put this condition in terms of $\alpha,k,a$. Compare this with the one given in the answer to 18(c) found on p. 732.
3. Section 2.6: 3, 8, 10, 14

Due Monday, June 16, 2014.

Assignment 4

• Problems. Many answers are in the back of the book. Thus, in order to receive credit, you must show all necessary steps in solving the problem. Also, the integrals must be done by hand - no symbolics. Use Matlab or your favorite program to do any required plots. Include labels for the axes, the equation plotted, the problem number and your name.
1. Section 3.1: 7, 8, 12, 19, 21
2. Section 3.2: 5, 9, 23, 25, 31

Due Friday, June 20, 2014.

Assignment 5

• Problems. Many answers are in the back of the book. Thus, in order to receive credit, you must show all necessary steps in solving the problem. Also, the integrals must be done by hand - no symbolics. Use Matlab or your favorite program to do any required plots. Include labels for the axes, the equation plotted, the problem number and your name.
1. Section 3.3: 7, 14, 18, 27, 38
2. Section 3.4: 11, 15(a,b,c), 23
3. Consider the differential operator $L[y]=y''+p(t)y'+q(t)y$. If $L[y_1]=0$, Reduction of order requires that we find another solution of the form $y_2=vy_1$. Use Abel's theorem to show that $v'=Cy_1^{-2}\exp\big(-\int p(t)dt\big)$. Use this equation to solve problem 25 in section 3.4.
4. Section 3.5: 4, 9

Due Wednesday, June 25, 2014.

Assignment 6