HW #3, due Tuesday February 24
==============================

Three problems: two problems are from the book,
and one is given here.



Problem #1
==========
Section 5.9, problem 2(b).

Comments:
Use RK4 with h as specified in the  problem;
plot computed u1(t) and exact u1(t) in one figure,
then plot computed u2(t) and exact u2(t) in the second
figure.
As usual, submit the code and both figures.




Problem #2
==========
Section 5.9, problem 4(c).

Comments:
Convert the 3-rd order equation into a system;
then use RK4.
Plot computed y(t) and exact y(t) in one figure.




Problem #3
==========
The motion of a toy car is described by the following
system:


x'      = v cos(theta)
y'      = v sin(theta)
theta'  = v/L*tan(psi)
v'      = a - gamma*v

Here x(t), y(t) are coordinates of the car,
theta(t) is the angle the car makes with the x-axis,
v(t) is the velocity of the car.

Parameters are:
a is the acceleration at the moment 0, a = 100 cm/sec^2
L is the length of the car, L = 11.5 cm,
psi is the angle between the front wheels and the car's axis, psi = 1(rad)
gamma = 1/6 (1/sec)

Initial conditions:
x(0)     = 10     (cm)
y(0)     = 10     (cm)
theta(0) = 1      (rad)
v(0)     = 0      (cm/sec)


Start at t = 0, with time step = 0.05 and proceed until y(t)
becomes less than 9. Plot the trajectory of the car.