Chapter 0 Ordinary Differential Equations 7
Roots of Characteristic General Solution of Differential
Equation Equation
Real, distinct roots:m 1 =m 2 u(t)=c 1 tm^1 +c 2 tm^2
Real, double root:m 1 =m 2 u(t)=c 1 tm^1 +c 2 (lnt)tm^1
Conjugate complex roots: u(t)=c 1 tαcos(βlnt)+c 2 tαsin(βlnt)
m 1 =α+iβ,m 2 =α−iβ
Table 2 Solutions oft^2 ddt^22 u+ktdudt+pu= 0whereλ>0. The characteristic equation ism(m− 1 )+m−λ^2 =m^2 −λ^2 =0.
The roots arem=±λ, so the first case of Table 2 applies, and
u(t)=c 1 tλ+c 2 t−λ (20)is the general solution of Eq. (19).
Forthegenerallinearequation
d^2 u
dt^2 +k(t)du
dt+p(t)u=^0 ,any point wherek(t)orp(t)fails to be continuous is asingular pointof the
differential equation. At such a point, solutions may break down in various
ways.However,ift 0 is a singular point where both of the functions
(t−t 0 )k(t) and (t−t 0 )^2 p(t) (21)have Taylor series expansions, thent 0 is called aregular singular point.The
Cauchy–Euler equation is an example of an important differential equation
having a regular singular point (att 0 =0). The behavior of its solution near
that point provides a model for more general equations.
- Other equations
Other second-order equations may be solved by power series, by change of
variable to a kind already solved, or by sheer luck. For example, the equation
t^4 d(^2) u
dt^2
+λ^2 u= 0 , (22)
which occurs in the theory of beams, can be solved by the change of variables
t=^1 z, u(t)=^1 zv(z).