Homework 3, due September 25, 2013.

[goodwine]

Read all Chapter 3.

Do exercises 3.1, 3.5, 3.7, 3.11, 3.18 (2, 4 and 7 only), 3.23 and 3.29.

[goodwine]

Someone asked me:

Regarding question 3.11, I am curious as to whether you want us to complete a plot for each condition. That is, do you want us so solve the equation seven times under the seven conditions and plot the solution to each condition? I only ask because the phrasing in the book implies a single solution, which doesn't make sense in context of what the first part of the problem asks us to do.

You can solve it multiple times if you prefer, but there is a single solution. You can leave the coefficients of the differential equation unspecified and solve for the constants in the homogeneous part of the solution for unspecified initial conditions, and then use that one function to plot it. If you would prefer, you could solve it multiple times. If the problem doesn't specify a method, you can generally take whatever approach you feel is best.

[John Hollkamp]

Concerning problem 3.23, this is a second order, linear, inhomogeneous, variable coefficient problem. I know I can use the method of variation of parameters to solve this ODE, but I do not know how to find the homogeneous solutions for a 2nd order variable coefficient ODE. Can I just let x1 and x2 be two linearly independent homogeneous solutions and then leave the general solution in terms of x1 and x2?

[goodwine]

Concerning problem 3.23, this is a second order, linear, inhomogeneous, variable coefficient problem. I know I can use the method of variation of parameters to solve this ODE, but I do not know how to find the homogeneous solutions for a 2nd order variable coefficient ODE. Can I just let x1 and x2 be two linearly independent homogeneous solutions and then leave the general solution in terms of x1 and x2?

If it's not possible to find the homogeneous solutions, then the correct answer is to say that we have not covered any method to find the homogeneous solutions in this case and therefore you can't solve the problem.

[goodwine]

Someone asked me:

For problem 3.11, there is a graph displayed as an example of a solution to 3.11. Is that the graph of the solution for the coefficients provided (m=2, b=1, k=10), or slightly different ones? Because with my equations, I get the graph on page 117 with coefficients m=1, b=1, and k=10. I'm not sure if this discrepancy is due to my equations, or if the graph in the book is with different coefficients.

It is supposed to be the solution for the indicated parameter values (specifically m=2), but of course I could have made a mistake making the graphics. Check again and if you are sure you are right, I'll add it to the errata list. You can still basically answer the question because the answers are things like showing how the frequency changes, whether it decays faster or slower, etc.

While I really do appreciate your good eye for the error, the way not to answer the problem is just by plotting and changing the values. I'm not sure if this is what you are doing. You should look at the expression for the solution and tie back the coefficient values and then interpret the manner in which the solution will change.

[jnorby]

For 3.11, should we include the graphs generated in matlab in the homework to be turned in, or is the plotting just for us to verify our predictions?

[goodwine]

For 3.11, should we include the graphs generated in matlab in the homework to be turned in, or is the plotting just for us to verify our predictions?

If they are very close to your sketch, then you don't have to submit the plots. If they are different, then submit them and revise your prediction.