Homework assigned 12/6, due 12/15
Section 9.1
Problems 2b, 3b
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?
Section 9.2
Problems 1b,2a,2b
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.
Section 9.3
Problems 4,7
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.
Section 9.4
Problems 2b,6,7
For #2b, only compute up to x=1. Present the errors at x=.2, .4, .6,
.8, and 1
Nonlinearities make life interesting
Consider the coupled system:
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?