For the following problem, assume that y(x) is a real-valued function.

A) For y' = a y, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points. How does your answer depend on the constant a?

B) For y' = y^2, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points.

C) For y' = -y^2, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points.

D) For y' = y^3, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points.

E) For y' = -y^3, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points.

F) For y' = a + y^2, find all the fixed points of the ODE. Find whether points initially near the fixed points move toward the fixed point or away from the fixed points. How does your answer depend on the constant a?

For #1b, only compute up to x=1.

For #2a, present one graph that contains of y(x,h=.5), y(x,h=.25), and y(x,h=.125). Present another graph that contains the errors err(x,h=.5), err(x,h=.25), and err(x,h=.125).

For #2b, present the same types of graphs that you presented for #2a.

For #1, you refer to table #1 in section 9.2, which is for the problem Y' = -Y, Y(0)=1. On the other hand, in section 9.3, you have equation (9.41) which tells you that the error should be bounded by h/2(exp(b)-1). In fact, you can compute that upper bound for various values of h and b. Do this and make sure that the errors in table 1 really are bounded by h/2(exp(b)-1).

For #4, you're asked to do the same thing: compute the upper bound of equation (9.38) and make sure that it bounds the errors that you find computationally. For this problem, _also_ find the ratios of the errors and make sure that they're behaving as you'd expect them to behave.

For #7, find the asymptotic bound at points x=2,3,4,5,6 and compare the bound to the numerical error you find in computing the solution using Euler's method.

For #2b, only compute up to x=1. Present the errors at x=.2, .4, .6, .8, and 1

dx/dt = y - (x^2+y^2-1) x

dy/dt = -x - (x^2 + y^2- 1) y

1) Find the fixed points (the steady-state solutions). Find analytically using pen and paper or maple.

2) Study the system numerically using second-order Runge-Kutta to do the time-stepping. First, give graphs demonstrating the behavior near the fixed point(s) (one graph per fixed point). Second, give a graph demonstrating more global behavior.

Consider the coupled system:

dx/dt = y + (x^2+y^2-1) x

dy/dt = -x + (x^2 + y^2- 1) y

Repeat the above three problems.

Consider the coupled system:

dx/dt = y + a (x^2+y^2-1) x

dy/dt = -x + a (x^2 + y^2- 1) y

where a is some fixed real number.

1) Find the fixed points (the steady-state solutions).

2) Find the behavior of solutions that start near the fixed points.

3) Can you find a geometrical explanation for the behavior of the solutions for different values of a?