Up: Fall 1999

EML 5060 Homework Set 3 Fall 1999

Page HW Class Topic
6 1.15 1.14 Notations
6 1.16   Notations
6 1.19   Notations
6 1.23 1.21 Notations
6 1.27   Solution by inspection
23 3.36 3.39 Separation of variables
23 3.43 3.42 Separation of variables
23 3.51 3.50 Homogeneous equations
23 3.53   Homogeneous equations
33 4.31 4.32 Exact equations
33 4.33   Exact equations
33 4.38   Exact equations
41 5.47 5.34 Linear equations
41 5.39 5.38 Bernoulli equations
63 6.34   Radioactive decay
64 6.58   Air resistance
66 6.74   RC circuit
81 8.27 8.18 Constant coefficient equations
81 8.30 8.19 Constant coefficient equations
81 8.38 8.21 Constant coefficient equations
86 9.18   Constant coefficient equations
86 9.19   Constant coefficient equations
96 10.44 10.45 Constant coefficient equations
96 10.46   Constant coefficient equations
96 10.48 10.47 Constant coefficient equations
102 11.12 11.10 Constant coefficient equations
102 11.15   Constant coefficient equations
102 11.28 11.25 Constant coefficient equations
107 12.12 12.11 Constant coefficient equations
122 13.39   Spring mass system
123 13.52   RCL circuit
124 13.71   Unsteady buoyancy
198 22.20 22.12 Solve as a system (required)

Note: make a graph of the solution for each solved problem.

Also solve the 4 questions below:

1.
Solve the Cauchy equation

\begin{displaymath} x^2 y''+ xy' - 4y = \ln x^2 \end{displaymath}

by taking $u=\ln \vert x\vert$ as the new independent variable. To eliminate x, use the chain rule of differentiation as in

\begin{displaymath} y' \equiv {dy\over dx} = {dy\over du} {du\over dx} = {dy\over du} {1\over x}, \end{displaymath}

and once more to find y'' in terms of dy/du and d2y/du2. Please do not indicate dy/du also by y'! Solution:

\begin{displaymath} y = -{\textstyle{1\over 2}} \ln x + A x^2 + B x^{-2} \end{displaymath}

2.
Solve the aerodynamically damped spring-mass system

\begin{displaymath} \ddot y + \left(\dot y\right)^2 + y = 0\end{displaymath}

by taking y as the independent variable and $\dot y$ as the dependent variable. To eliminate the remaining dt, (in $\ddot y = d\dot y/dt$), use the chain rule of differentiation. Solution:

\begin{displaymath} \dot y^2 = -y + {\textstyle{1\over 2}} + C_0 e^{-2y} \mbox{,... ...} t = \pm \int {dy\over \sqrt{- y + {1\over 2} + C_0 e^{-2y}}} \end{displaymath}

3.
Solve the motion of a falling body with aerodynamic drag:

\begin{displaymath} \ddot x + \left(\dot x\right)^2 = 1. \end{displaymath}

Solution:

\begin{displaymath} \dot x = {Ce^{2t} -1 \over C e^{2t}+1} \quad x = \ln\vert Ce^{2t}+1\vert - t + D \end{displaymath}

4.
Solve the equation for the streamfunction in a Stokes boundary layer:

y'' + 2xy' - 2y = 0.

Note that y=x is one solution. Solution:

\begin{displaymath} y = C_0 x + C_1 x \int {e^{-x^2}\over x^2} dx \end{displaymath}

Up: Fall 1999