1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

4 Chapter 0 Ordinary Differential Equations

Roots of Characteristic General Solution of Differential

Equation Equation

Real, distinct:m 1 =m 2 u(t)=c 1 em^1 t+c 2 em^2 t

Real, double:m 1 =m 2 u(t)=c 1 em^1 t+c 2 tem^1 t

Conjugate complex: u(t)=c 1 eαtcos(βt)+c 2 eαtsin(βt)

m 1 =α+iβ,m 2 =α−iβ

Table 1 Solutions ofddt^2 u 2 +kdudt+pu= 0

cases, a pair of complex conjugate rootsm=α±iβleads to a pair of complex
solutions

eαteiβt, eαte−iβt (10)

that can be traded for the pair of real solutions

eαtcos(βt), eαtsin(βt). (11)

We include two important examples. First, consider the differential equation

d^2 u

dt^2

+λ^2 u= 0 , (12)

whereλis constant. The characteristic equation ism^2 +λ^2 =0, with roots
m=±iλ. The third case of Table 1 applies ifλ=0; the general solution of the
differential equation is

u(t)=c 1 cos(λt)+c 2 sin(λt). (13)

Second, consider the similar differential equation

d^2 u

dt^2

−λ^2 u= 0. (14)

The characteristic equation now ism^2 −λ^2 =0, with rootsm=±λ.Ifλ=0,
the first case of Table 1 applies, and the general solution is

u(t)=c 1 eλt+c 2 e−λt. (15)

It is sometimes helpful to write the solution in another form. The hyperbolic
sine and cosine are defined by

sinh(A)=

1

2

(

eA−e−A

)

, cosh(A)=

1

2

(

eA+e−A

)

. (16)

Thus, sinh(λt)and cosh(λt)are linear combinations ofeλtande−λt.Bythe
Principle of Superposition, they too are solutions of Eq. (14). The Wronskian