Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
Ordinary and singular points of a di€erential equation
We shall concentrate on the linear second-order di€erential equation of the form
d 2 y dy
‡ P…x†
dx2 dx ‡ Q…x†y ˆ 0
…2:24†
which plays a very important part in physical problems, and introduce certain
de®nitions and state (without proofs) some important results applicable to equations
of this type. With some small modi®cations, these are applicable to linear
equation of any order. If both the functions P and Q can be expanded in Taylor
series in the neighborhood of x ˆ , then Eq. (2.24) is said to possess an ordinary
point at x ˆ . But when either of the functions P or Q does not possess a Taylor
series in the neighborhood of x ˆ , Eq. (2.24) is said to have a singular point at
x ˆ . If
P ˆ …x†=…x † and Q ˆ …x†=…x † 2
and …x† and …x† can be expanded in Taylor series near x ˆ . In such cases,
x ˆ is a singular point but the singularity is said to be regular.
Frobenius and Fuchs showed that:
Frobenius and Fuchs theorem
(1) If P…x† and Q…x† are regular at x ˆ , then the di€erential equation (2.24)
possesses two distinct solutions of the form
y ˆ X1
ˆ0
a …x † …a 0 6ˆ 0†: …2:25†
(2) If P…x† and Q…x† are singular at x ˆ , but …x †P…x† and …x † 2 Q…x†
are regular at x ˆ , then there is at least one solution of the di€erential
equation (2.24) of the form
y ˆ X1
ˆ0
a …x † ‡ …a 0 6ˆ 0†; …2:26†
where is some constant, which is valid for jx j