Homework 12

CHE 116: Numerical Methods and Statistics

Prof. Andrew White

Version 1.1 (4/9/2015)

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1. Rearranging ODEs (4 Points)

Convert the following ODEs into standard form. If it is a second-order, make sure you specify both dimensions.

1. $\frac{dx}{dt} - t = 2$

2. $\frac{dx}{dt} = -rx$

3. $\frac{d^2x}{dt^2} + k x = \sin t$

4. $\frac{d^2x}{dt^2} + b \frac{dx}{dt} + k x = 0$

2. Solve This ODE (6 Points)

Solve the tank problem from lecture:

$$\frac{dV}{dt} + k_1 \sqrt{V} = k_2$$

and use $k_1 = 1$, $k_2 = 4$. Plot the solution from $t=0$ to $t=25$ with the initial condition that $V_0 = 10$

3. Solve This Other ODE (6 Points + 2 EC)

Solve a forced, dampened ODE governed by this equation:

$$\frac{d^2x}{dt^2} + \frac{1}{10}\frac{dx}{dt} + k x = 5e^{-c / t}$$

using $c=0.25$ and $k=2.5$. For 2 points extra credit, instead create an interactive widget. Use an initial condition of position 0.0 and velocity 1.0.

4. Linear Algebra (3 Points)

Given that:

$$\mathbf{A} = \left[\begin{array}{lcr} 7 & 0 & 1\\ 1 & 3 & 4\\ 4 & 5 & 2\\ \end{array}\right]$$$$\mathbf{b} = \left[\begin{array}{lcr} 4\\ 13\\ -6\\ \end{array}\right]$$

Answer the following problems:

1. What is $\mathbf{A}\mathbf{b}$?

2. What is the largest eigenvector of $\mathbf{A}$?

3. Solve $\mathbf{A}\mathbf{x} = \mathbf{b}$