Computer Algebra Recipes

4.1. FIRST-ORDER MODELS 163
μ
tg
v := tanh V
V
To determine y(t), a second ODE is formed by equating dy=dt to v. To make
the ¯nal expression appear in the desired form, the convert command is used
to reexpress the hyperbolic tangent in terms of the hyperbolic sine and cosine.
> ode2:=diff(y(t),t)=convert(v,sincos);
ode2 := d
μ
tg
sinh V
V
y(t) = μ
dt tg
cosh
V
Analytically solving ode2 for y(t), subject to the initial condition y(0) = 0,
> dsolve(fode2,y(0)=0g,y(t));
V
y(t) =
2 μ μ
tg
ln cosh
V
g
produces an output that is identical in structure to equation (4.4), thus completing
Vectoria's task.
PROBLEMS:
Problem 4-15: Return velocity
A ball of unit mass is thrown vertically upward near the earth's surface with an
initial speed v(0) = U. Assuming that Newton's law of air resistance prevails,
show that after the ball rises to its maximum height and begins to fall it passes
its initial position with a velocity v = (UV)=( p U 2 + V 2 ); where V is the
terminal velocity. Hint: Reexpress the acceleration as
dv
dt =
μ μ μ
dv dx
dv(x)
= v(x)
dx dt
dx
and note that at the maximum height the speed is zero.
Problem 4-16: A Riccati equation
Consider the following Riccati equation:
y 0 (x)+y(x)=x + ay(x) 2 + b =0;
with the initial condition y(0) = A, wherea, b, andAare real constants.
(a) Analytically solve the ODE. Does the answer depend on the value of A?
(b) What does the general solution look like if no initial condition is speci¯ed?
Explain what happens mathematically when y(0) = A is imposed.
(c) Taking a =5andb =2,plotthesolutionfory(0) = A over the range
x = 0 to 2, using the plot option view=[0..2,-5..5].
(d) Explain the origin of the singular points in the graph in terms of the
behavior of Bessel functions.