Bill Goodwine's Courses

Reading: all of Chapter 3.

Problems: 3.1, 3.2, 3.5, 3.6, 3.7, 3.10, 3.11, 3.18 (2, 4 and 7 only), 3.23 and 3.26.

Professor, in selecting problem 3.23 for this homework assignment -
{ Determine the general solution to d^2x/dt^2 - (2*t) dx/dt + x = sec(t) }
did you realize that, as the chapter summary indicates:

"For ordinary, second-order, linear, variable-coefficient, inhomogeneous differential equations, the method of variation of parameters works. However, two linearly independent homogeneous solutions are required for the method, and at least at this point, you do not have any method to find them."?

In light of this, are we supposed to determine the solution method for linear, variable-coefficient, homogeneous diff eqs ourselves, or did you plan on teaching that to us on Monday?

Can we solve problem 3.23 using Variation of Parameters? If we need the homogeneous solution in order to find the particular solution, how can we find the homogeneous solution if has variable coefficients?

Also, for problem 26, did you just want us to analyze the second order equations from problem 1.8 or all of the problems, regardless of order?

sprender wrote:Also, for problem 26, did you just want us to analyze the second order equations from problem 1.8 or all of the problems, regardless of order?

I could be wrong (and Professor Goodwine can confirm/deny), but I believe we are just supposed to say which of the given solution methods work for each Diff Eq in problem 1.8 regardless of order since the solution methods are more or less independent of order.

e.g.
if the equation were an ordinary, first order, linear, variable coefficient, homogeneous differential equation that is separable, then the answer might be something like:
3 (VoP), 4 (separable), 5 (exact)

Hope that helps until Prof Goodwine can give a better answer!

runkle89 wrote:Professor, in selecting problem 3.23 for this homework assignment -
{ Determine the general solution to d^2x/dt^2 - (2*t) dx/dt + x = sec(t) }
did you realize that, as the chapter summary indicates:

"For ordinary, second-order, linear, variable-coefficient, inhomogeneous differential equations, the method of variation of parameters works. However, two linearly independent homogeneous solutions are required for the method, and at least at this point, you do not have any method to find them."?

In light of this, are we supposed to determine the solution method for linear, variable-coefficient, homogeneous diff eqs ourselves, or did you plan on teaching that to us on Monday?

If we have not yet covered a solution method that works for it, then the correct answer is "we have not yet covered a solution method that works for this equation."

sprender wrote:Also, for problem 26, did you just want us to analyze the second order equations from problem 1.8 or all of the problems, regardless of order?

The problem says "for each of the second-order equations listed in problem 1.8" so you only need to do the second order ones.

runkle89 wrote:

sprender wrote:Also, for problem 26, did you just want us to analyze the second order equations from problem 1.8 or all of the problems, regardless of order?

I could be wrong (and Professor Goodwine can confirm/deny), but I believe we are just supposed to say which of the given solution methods work for each Diff Eq in problem 1.8 regardless of order since the solution methods are more or less independent of order.

e.g.
if the equation were an ordinary, first order, linear, variable coefficient, homogeneous differential equation that is separable, then the answer might be something like:
3 (VoP), 4 (separable), 5 (exact)

Hope that helps until Prof Goodwine can give a better answer!

You definitely can do that and it would not be a bad exercise, but the problem only is for the second order ones.

sprender wrote:Can we solve problem 3.23 using Variation of Parameters? If we need the homogeneous solution in order to find the particular solution, how can we find the homogeneous solution if has variable coefficients?

You are right. So the correct answer is that we have not yet covered a method to solve it.

For problem 3.10 in the diffeq homework set, I graphed each of the 4 equations, but none of the equations seem to change in slope. My understanding is that equation 2 and 3 should show a change in slope due to the opposing signs. However, graphically the first two only decay exponentially and the second two grow exponentially. I've talked to a couple of people about it, and they are running into the same problem.

Make sure that you limit your plot to, for example, t=0 to t=5. I think you'll see a slope change in that case.