Applications of Linear Equations with Constant Coefficients 303[yp) + (b + c + f)y~' + bcy~]/(ef) =
c(y~ - au+ by1)/ f - (d + e)[y{ + (b + c + f)y~ + bcy1 - (c + f)au]/(ef).
Multiplying by ef and rearranging, we see that y 1 satisfies the third-order,
linear differential equation(36) yi
3
) + (b+ c+ d + e + f)y{ + [b(c + d + e) + cd + f(d + e)]y~ + bcdy 1 =[cd + f(d + e)]au.
Since the volume of each compartment remains constant, the sum of the input
rates per volume into a compartment must equal the sum of the output rates
per volume from the compartment. So the rates per volume a, b, c, d, e, andf must also satisfy the equations
a+f =b
b+e=c+f
c=d+e(for compartment Y 1 )
(for compartment Y 2 )
(for compartment Y3)Exercise 5. Find the concentration of a substance in each of the compart-
ments of Figure 6. 10 as a function of time, if a = 1 min-^1 , b = 4 min-1, c =
7 min-1, d = 1min-1,e=6 min-^1 , f = 3 min-^1 , y 1 (0) = .25, Y2(0) = .4 ,
and y3 = .7. (HINT: Use POLYRTS or your computer software to find the
roots of the auxiliary equation associated with equation (36). Write the gen-
eral solution of equation (36). Then find Y2 and y 3. And finally, satisfy the
initial conditions.)
Beams and Columns Beams and columns are common structural ele-
ments used in t he construction of airplanes, bridges, buildings, and ships. Be-
cause of their importance in construction, beams and columns were studied ex-
tensively by ancient Greek and Roman a rchitects. Galileo and Coulomb both
made contributions to the early theory of the deflection of beams and bending
of columns. However , our modern engineering theory regarding beams and
columns h as its origin during the eighteenth century in studies conducted by
Euler and t he Bernoullis.
An ideal beam is a lo ng, slender, nonv.ertical rod which is supported at
one or both ends and which is usually subject to external forces acting on
it. The external forces may act at any point or points along the length of
the beam and thereby cause a displacement of the beam from its unloaded
position. The displacement of a beam from its unlo aded position is described
by a fourth-order ordinary differential equation.
Consider a horizontally placed beam which does not rest on an elastic foun-
dation as shown in Figure 6.11. Let x denote the horizontal distance from the