Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
where k is a constant of integration (in electric circuits, C is used for capacitance).
Given E this equation can be solved for I…t†. If the voltage E is constant, we
obtain
I…t† ˆ1
L eRt=L
E L
R eRt=L
Regardless of the value of k, we see that
I…t† !E=R as t !1:
Setting t ˆ 0 in the solution, we ®nd
k ˆ I…0†E=R:
‡ ke Rt=L ˆ E
R ‡ keRt=L :
Bernoulli's equation
Bernoulli's equation is a non-linear ®rst-order equation that occurs occasionally
in physical problems:
dy
dx ‡ f …x†y ˆ g…x†yn ;
…2:14†
where n is not necessarily integer.
This equation can be made linear by the substitution w ˆ y a with suitably
chosen. We ®nd this can be achieved if ˆ 1 n:
w ˆ y 1n or y ˆ w 1=…1n† :
This converts Bernoulli's equation into
dw
‡…1 n† f …x†w ˆ…1 n†g…x†;
dx
which can be made exact using the integrating factor exp… R …1 n† f …x†dx†.
Second-order equations with constant coe
cients
The general form of the nth-order linear di€erential equation with constant coef-
®cients is
d n y
dx n ‡ p d n1 y
1
dx n1 ‡‡p dy
n1
dx ‡ p ny ˆ…D n ‡ p 1 D n1 ‡‡p n1 D ‡ p n †y ˆ f …x†;
where p 1 ; p 2 ; ... are constants, f …x† is some function of x, and D d=dx. If
f …x† ˆ0, the equation is called homogeneous; otherwise it is called a non-homogeneous
equation. It is important to note that the symbol D is meaningless unless
applied to a function of x and is therefore not a mathematical quantity in the
usual sense. D is an operator.
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