Mathematical Methods for Physicists: A concise introduction - Site Map

SECOND-ORDER EQUATIONS WITH CONSTANT COEFFICIENTS
This is called the auxiliary (or characteristic) equation of Eq. (2.16). Solving it
gives
p 1 ˆ a ‡
p
p
a 2 4b a a 2 4b
; p
2
2 ˆ : …2:18†
2
We now distinguish between the cases in which the roots are real and distinct,
complex or coincident.
(i) Real and distinct roots (a 2 4b > 0†
In this case, we have two independent solutions y 1 ˆ e p1t ; y 2 ˆ e p2t and the general
solution of Eq. (2.16) is a linear combination of these two:
y ˆ Ae p1t ‡ Be p2t ;
…2:19†
where A and B are constants.
Example 2.9
Solve the equation …D 2 2D 3†y ˆ 0, given that y ˆ 1 and y 0 ˆ dy=dx ˆ 2
when t ˆ 0.
Solution: The auxiliary equation is p 2 2p 3 ˆ 0, from which we ®nd p ˆ1
or p ˆ 3. Hence the general solution is
y ˆ Ae t ‡ Be 3t :
The constants A and B can be determined by the boundary conditions at t ˆ 0.
Since y ˆ 1 when t ˆ 0, we have
1 ˆ A ‡ B:
Now
y 0 ˆAe t ‡ 3Be 3t
and since y 0 ˆ 2 when t ˆ 0, we have 2 ˆA ‡ 3B. Hence
A ˆ 1=4; B ˆ 3=4
and the solution is
4y ˆ e t ‡ 3e 3t :
(ii) Complex roots …a 2 4b < 0†
If the roots p 1 , p 2 of the auxiliary equation are imaginary, the solution given by
Eq. (2.18) is still correct. In order to give the solutions in terms of real quantities,
we can use pthe Euler relations to express the exponentials. If we let
r ˆa=2; is ˆ
a 2 4b=2, then
e p 1t ˆ e rt e ist ˆ e rt ‰cos st ‡ i sin stŠ;
e p 2t ˆ e rt e ist ˆ e rt ‰cos st i sin stŠ
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