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∂
g z () = z g z ()
32/ z ∂ 5 2/

where

2
2
1
g z ( ) =− 4 ∫ ∞ dxx ln( − zexp(− x ))
52/ π 0
and z, for physical reasons, always remains in the interval 0 < z ≤ 1.
a. Plot g 5/2 (z) as a function of z over the interval 0 < z ≤ 1.
b. Plot g 3/2 (z) over the same interval and find its maximum value.
c. As n increases or T decreases, the second term on the rhs of the
population equation keeps adjusting the value of z so that the
two terms on the RHS cancel each other, thus keeping n condensate =
0. However, at some point, z reaches the value 1, which is its
maximum value and the second term on the RHS cannot increase
further. At this point, n condensate starts building up with any increase
in the total number density. The value of the total density at
which this starts happening is called the threshold value for the
condensate formation. Prove that this threshold is given by:
λ =
n threshold 3 2 612. .
T

4.7 Numerical Solutions of Ordinary Differential Equations

Ordinary linear differential equations are of the form:

n
dy d n−1 y dy
at() + a t () +…+ at() + at y() = u t() (4.24)
n n n−1 n−1 1 0
dt dt dt
The a’s are called the coefficients and u(t) is called the source (or input) term.
Ordinary differential equations (ODEs) show up in many problems of elec-
trical engineering, particularly in circuit problems where, depending on the
circuit element, the potential across it may depend on the deposited charge,
the current (which is the time derivative of the charge), or the derivative of
the current (i.e., the second time derivative of the charge); that is, in the same
equation, we may have a function and its first- and second-order derivatives.
To focus this discussion, let us start by writing the potential difference across
the passive elements of circuit theory. Specifically, the voltage drops across a
resistor, capacitor, or inductor are given as follows:

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