Page 264 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 10 ORDINARY DIFFERENTIAL EQUATIONS. PART I 24 1

Problems

Answers to the following problems are found in the folder "Ch. 10 (ODE)" in the
"Problems & Solutions" folder on the CD.

1. A function is described by the differential equation dyldt = 1 - t fi .
Calculate y for t = 0 to t =5, in increments of 0.1.

2. A function is described by the differential equation
dy - 1 - 2x2 /(1+ x2)
-
-
dx l+x2
Calculate y for x = 0 to x = 6.

3. A function is described by the differential equation

y - arctan(x1 +
dx l+x

Calculate y for x = 0 to x = 2.5. Adjust the magnitude of Ax for different
parts of the calculation, as appropriate.

4. Trajectory I. Consider the motion of a projectile that is fired from a cannon.
The initial velocity of the projectile is vo and the angle of elevation of the
cannon is B degrees. If air resistance is neglected, the velocity component of
the projectile in the x direction (x') is vo cos 8 and the component in the y
direction is vo sin B-gt. Use Euler's method to calculate the trajectory of the
projectile. For the calculation, assume that the projectile is a shell from a
122-mm field howitzer, for which the muzzle velocity is 560 ds. (Getting
started: create five columns, as follows: t, XI, y', x, y. Calculate x and y, the
coordinates of distance traveled, from, e.g., x(+~ = xt + x,'At.) Verify that the
maximum range attainable with a given muzzle velocity occurs when B =
45".

5. Trajectory 11. Without air resistance, the projectile should strike the earth
with the same yl that it had when it left the muzzle of the cannon. Because of
accumulated errors when using the Euler method, you will find that this is
not true. Repeat the calculation of problem number 1 using RK4.

6. Trajectory III. To produce a more accurate estimate of a trajectory, air drag
should be taken into account. For speeds of objects such as baseballs or
cannonballs, air drag can be taken to be proportional to the square of the