Bill Goodwine's Courses

Reading: for chapter 12 you are responsible for all the material except sections 4 and 6.

Reading: Chapter 13, sections 1 and 2 only.

Exercise: Consider the equation given in number 2 of Exercise 13.2. For that equation

1. Find the equilibrium points.
2. By using a Taylor series expansion for the nonlinear term(s), determine a linear differential equation that approximates the nonlinear equation near each equilibrium point. Find the solution for each linear equation. Which ones are stable and which are unstable?
3. Write a C or FORTRAN program that uses 4th order Runge-Kutta to solve the nonlinear equation.
4. Near each equilibrium point, in the phase plane plot the numerical solution to the nonlinear equation for several initial conditions. On the same graph, plot the solution to the linear approximation. For the linear equation, you can either use the solution you determined for the previous part, or you could find a numerical solution. Verify that as the solution moves farther away from the equilibrium point, the solution to the linear equation becomes a worse approximation to the nonlinear solution. Basically, make plots like Figures 13.8-13.13.

what initial conditions would you like us to use to do the runge-kutta?

cdiberna wrote:what initial conditions would you like us to use to do the runge-kutta?

Just pick your own. The point is to test how accurate the linear approximation is close to and not so close to the equilibrium point.

Should we be plotting both the non-linear solution and the linear approximation for each initial condition on one graph? It seems like it could get a bit cluttered.

kcollin8 wrote:Should we be plotting both the non-linear solution and the linear approximation for each initial condition on one graph? It seems like it could get a bit cluttered.

You can decide for yourself how to most effectively communicate the relationship between the two solutions. Don't waste a lot of paper, but don't submit something indecipherable in the interest of saving paper.

I'm a little bit foggy on the definition of stable and unstable solutions. Should we just use the guidelines on page 446? If so, are those guidelines referring to the eigenvalues from the homogeneous solution? Thanks so much.

vsteger wrote:I'm a little bit foggy on the definition of stable and unstable solutions. Should we just use the guidelines on page 446? If so, are those guidelines referring to the eigenvalues from the homogeneous solution? Thanks so much.

Where we are at this point: stable means the solution converges to a fixed value. Unstable means it blows up.

Professor,
We are having issues with part 4. When solving the numerical solution to the nonlinear equation for several initial conditions, what do we need to change in our code to change the equilibrium point? Every time we plot it in the phase plane, we get a plot around zero.

Thank you.

dwolf3 wrote:Professor,
We are having issues with part 4. When solving the numerical solution to the nonlinear equation for several initial conditions, what do we need to change in our code to change the equilibrium point? Every time we plot it in the phase plane, we get a plot around zero.

Thank you.

The equilibrium point is a feature of the nonlinear differential equation itself, so you don't need to change anything. You compute the linear approximation by hand, so that won't change in your code either. All you change are the initial conditions. Near the equilibrium, the solution to the linear equation should be a good approximation to the solution to the nonlinear equation.

In the book, for equation 13.7, why do you add a square root of 2 to the end of the equation?

i think the square root of two is the particular solution which added to the homogeneous solution gets you the full solution

joneill5 wrote:In the book, for equation 13.7, why do you add a square root of 2 to the end of the equation?

It's the particular solution.