Page HW Class Topic

6 1.18 1.14 Notations

6 1.19 Notations

6 1.23 1.21 Notations

6 1.25 Solution by inspection

23 3.38 3.39 Separation of variables

23 3.44 3.42 Separation of variables

23 3.48 3.50 Homogeneous equations

23 3.54 Homogeneous equations

33 4.29 4.32 Exact equations

33 4.35 Exact equations

42 5.52 5.34 Linear equations

42 5.53 5.38 Bernoulli equations

63 6.46 Heat conduction

64 6.63 Air resistance

66 6.87 Bacteria growth

81 8.29 8.18 Constant coefficient equations

81 8.40 8.19 Constant coefficient equations

81 8.41 8.21 Constant coefficient equations

86 9.17 Constant coefficient equations

86 9.20 Constant coefficient equations

96 10.46 10.45 Forcing

96 10.52 10.47 Forcing

102 11.13 11.10 Constant coefficient equations

102 11.14 Constant coefficient equations

102 11.29 11.25 Constant coefficient equations

107 12.12 12.11 Constant coefficient equations

122 13.36 Spring mass system

123 13.53 RCL circuit

124 13.77 Unsteady buoyancy

197 22.21 22.12 Solve as a system (required) 4|c| Solve the 4 questions below

Note: make a graph of the solution for each solved problem. If the equation has multiple solutions, assume that the value of the unknown at zero is unity, and all its derivatives are zero at zero.

Also make the following four questions (over):

1.

Solve the Cauchy equation

$$x^2 y''+ xy' - 4y = \ln x^2$$

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

$$y' \equiv {dy\over dx} = {dy\over du} {du\over dx} = {dy\over du} {1\over x},$$

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

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

2.

Solve the aerodynamically damped spring-mass system

$$\ddot y + \left(\dot y\right)^2 + y = 0$$

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:

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

3.

Solve the motion of a falling body with aerodynamic drag:

$$\ddot x + \left(\dot x\right)^2 = 1.$$

Solution:

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

4.

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

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

Note that y=x is one solution. Solution:

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