1000 Solved Problems in Modern Physics

1.1 Basic Concepts and Formulae 19

First step: Write down the corresponding auxiliary equation

Dn+p 1 Dn−^1 +p 2 Dn−^2 +···+pn= 0

Second step: Solve completely the auxiliary equation.
Third step: From the roots of the auxiliary equation, write down the correspond-
ing particular solutions of the differential equation as follows

Auxiliary equation Differential equation

(a) Each distinct real rootr 1 Gives a particular solutioner^1 x

(b) Each distinct pair of imaginary

rootsa±bi

Gives two particular solutions

eaxcosbx,eaxsinbx

(c) A multiple root occurringstimes Givessparticular solutions obtained by

multiplying the particular solutions

(a)or(b)by1,x,x^2 ,...,xn−^1

Fourth step: Multiple each of thenindependent solutions by an arbitrary constant
and add the results. This gives the complete solution.

Type II

(I)

dny

dxn

+P 1

dn−^1 y

dxn−^1

+P 2

dn−^2 y

dxn−^2

+···+Pny=X

whereXis a function ofxalone, or constant, andP 1 ,P 2 ,...Pnare constants.
WhenX=0, (I) reduces to (A) Type I.

(J)

dny

dxn

+P 1

dn−^1 y

dxn−^1

+P 2

dn−^2 y

dxn−^2

+···+Pny= 0

The complete solution of (J) is called the complementary function of (I).
Letube the complete solution of (J), i.e. the complementary function of (I), and
vany particular solution of (I). Then

dnv

dxn

+P 1

dn−^1 v

dxn−^1

+P 2

dn−^2 v

dxn−^2

+···+Pnv=X

and

dnu

dxn

+P 1

dn−^1 u

dxn−^1

+P 2

dn−^2 u

dxn−^2

+···+Pnu= 0

Adding we get

dn(u+v)

dxn

+P 1

dn−^1 (u+v)

dxn−^1

+P 2

dn−^2 (u+v)

dxn−^2

+···+Pn(u+v)=X

showing thatu+vis a solution ofI.