Math 325: Differential Equations	Assignment 3		Fall 1997

Names (up to 3 students):

Section:

Initialization

<<Graphics`PlotField`

Problem 1

Using eigenvalues and eigenvectors,  express the general solution of the system X'[t] == A.X[t] in terms of real-valued functions for each of the matrices A below. Check that your solutions are correct by comparing X'[t] and A.X[t]. Superimpose several plots for each using ParametricPlot3D[] or ParametricPlot3D[].

a)  A = {{ 3, 2},
       {-1, 1}};

b)  A = {{1,-1, 4},
       {3, 2,-1},
       {2, 1,-1}};
       
c)  A = {{-14,18, 8},
       { -6, 8, 2},
       { -5, 7, 3}};

Solution

Problem 2

For each of the matrices A below, find the fundamental matrix Phi[t] such that Phi[0] is the identity matrix for the system X'[t] == A.X[t]. Verify that Phi'[t]==A.Phi[t]. Then use Phi[t] to find a solution satisfying the given initial condition.

a)  A = {{-7,-8, 2},
        { 9,11,-3},
        { 9, 8, 0}};
  X[0]=={3,-1,1};

b)  A = {{-4,-2,-1},
        { 2, 6,-3},
        { 2, 4,-1}};
  X[0]=={1,1,-2};
 
c)  A = {{  5, 3,-1},
        {-16,-6,-1},
        { -4,-3, 2}};
  X[0]=={1,1,1};

Solution

Problem 3

Use the method of variation of parameters to find the general solution of the system X'[t] == A.X[t] + G[t] for each of the matrices A and vector functions G[t] below. Check that your solution is correct by comparing X'[t] and A.X[t]+G[t]. Then find the solution that satisfies the given initial condition.

a) (see Problem 2a)
   A = {{-7,-8, 2},
        { 9,11,-3},
        { 9, 8, 0}};
  G[t_] = {t^2,t,1};
  X[0]=={3,-1,1};

b) (see Problem 2b)
   A = {{-4,-2,-1},
        { 2, 6,-3},
        { 2, 4,-1}};
  G[t_] = {1,Cos[t],Sin[t]};
  X[0]=={1,1,-2};
 
c) (see Problem 2c)
   A = {{  5, 3,-1},
        {-16,-6,-1},
        { -4,-3, 2}};
  G[t_] = {E^t,1,E^(-t)};
  X[0]=={1,1,1};

Solution

Problem 4

Consider the autonomous system:
    x'[t] == 1 - x[t]y[t]
    y'[t] == x[t] - y[t]^3
a)  Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point.
b)  Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) (this makes the arrows bigger near the critical points) and Frame->True. Use the vector field plot to choose a representative sample of several initial points {x0,y0}. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t.

Solution

Problem 5

In this problem we consider the equations describing the motion of a pendulum, both damped and undamped. For each situation below convert the equation into a system of equations in x[t] = theta[t] and y[t] = theta'[t]. Plot the corresponding vector field. Use NDSolve[] to solve this system and plot on a single graph the trajectories corresponding to the given initial velocities b. Based on these plots, describe what the pendulum is doing in each case. Experiment to see if you can find a value for the initial velocity b such that the trajectory approaches the unstable equilibrium point {Pi,0} corresponding to the pendulum being stationary in an upright position.

a) The undamped pendulum:

    theta''[t] + Sin[theta] == 0,
    theta[0] == 0, theta'[0] == b

The initial velocities are b = 0.5 to 3.0 in steps of 0.5.

b)  The damped pendulum:

    theta''[t] + 0.5 theta'[t] + Sin[theta] == 0,
    theta[0] == 0, theta'[0] == b

The initial velocities are b = 0.5 to 6.0 in steps of 0.5.

Solution