Applications of the Initial Value Problem y' = f(x, y) ; y(c) = d 157The constant K must be chosen to satisfy the initial condition p(b) = -a/b.
Returning to equation (7), solving for K , and setting x =band p = -a/b, we
see that(9) K = (-a/b + J1 + a^2 /b^2 )/bR = (-a+ J a2 + b2)/bR+^1.
The path of the dog is the solution to the initial value problem consisting of
the differential equation (8) where K is given by equation (9) and the initial
condition y(b) = 0. Compute and graph the path of the dog if a = 200 ft,
b = 100 ft, Sr = 50 ft/sec, and (a) Sd = 40 ft/sec, (b) sd = 50 ft/sec, and
( c) sd = 75 ft/sec. In which case does the dog capture the rabbit? When and
where does the dog catch the rabbit?3.8 Chemical Reactions
Chemical equations indicate how molecules of substances combine or de-
compose to produce other substances. For example, the chemical equationstates that 2 molecules of hydrogen, H 2 , combine with 1 molecule of oxygen,
02, to form 2 molecules of water, H20. A general chemical equation has the
form
aA + bB + cC + · · · --) pP + qQ + · · ·
where a, b, c, ... , p, q, ... are positive integers. This equation indicates that a
molecules of the substance A , b molecules of the substance B, c molecules of
the substance C ,... react to form p molecules of the substance P, q molecules
of the substance Q,. ... The rate at which a substance is formed is called the
velocity of the reaction. In many cases the law of mass action stated below
applies in determining the velo city of reaction.
LAW OF MASS ACTIONIf the temperature is kept constant, the velocity of a chemical reaction
is proportional to the product of the concentrations of the reacting
substances.Recall from elementary chemistry that a mole is the number of grams of
a substance which is equal to the molecular weight of the substance. For
example, 1 mole of hydrogen, H2, has a weight of 2.016 grams, since the
atomic weight of the element hydrogen, H , is 1.008 grams; 1 mole of oxygen,
02 , has a weight of 32 grams, since the atomic weight of the element oxygen,
0, is 16 grams; and 1 mole of water, H 2 0, has a weight of 2(1.008) + 16 =
18.016 grams.