Page 33 - Advanced engineering mathematics

1.1 Terminology and Separable Equations 13

18
18–h
r
h

FIGURE 1.4 Draining a hemi-
spherical tank.

This is a separable differential equation which we write as

36h − h 2
π dh =−kA 2gdt.
h 1/2
Take g = 32 feet per second per second. The radius of the circular opening is 3 inches (or
1/4 feet), so its area is A = π/16. For water and an opening of this shape and size, experiment
gives k = 0.8. Therefore,
1 √
3/2
(36h 1/2 − h )dh =−(0.8) 64dt,
16
or
3/2
(36h 1/2 − h )dh =−0.4dt.
A routine integration gives us

2 2
24h 3/2 − h 5/2 =− t + c
5 5
with c as yet an arbitrary constant. Multiply by 5/2 to obtain

60h 3/2 − h 5/2 =−t + C
with C arbitrary. For the problem under consideration, the radius of the hemisphere is 18 feet, so
h(0) = 18. Therefore,
60(18) 3/2 − (18) 5/2 = C.
√
Then C = 2268 2, and
√
60h 3/2 − h 5/2 = 2268 2 − t.
√
The tank is empty when h = 0, and this occurs when t = 2268 2 seconds or about 53 minutes,
28 seconds. This is time it takes for the tank to drain.

These last three examples illustrate an important point. A differential equation or initial
value problem may be used to model and describe a process of interest. However, the process
usually occurs as something we observe and want to understand, not as a differential equation.
This must be derived, using whatever information and fundamental principles may apply (such
as laws of physics, chemistry, or economics), as well as the measurements we may take. We
saw this in Examples 1.5, 1.6, and 1.7. The solution of the differential equation or initial value
problem gives us a function that quantifies some part of the process and enables us to understand
its behavior in the hope of being able to predict future behavior or perhaps design a process
that better suits our purpose. This approach to the analysis of phenomena is called mathematical
modeling. We see it today in studies of global warming, ecological and financial systems, and
physical and biological processes.

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October 14, 2010 14:9 THM/NEIL Page-13 27410_01_ch01_p01-42