1000 Solved Problems in Modern Physics

18 1 Mathematical Physics

Type IV equations reducible to linear form

Some equations that are not linear can be reduced to the linear form by a suitable
substitution, for example

(A)

dy

dx

+Py=Qyn

whereP,Qare functions ofxalone, or constants. Equation (A) may be reduced to
the linear form (A), Type III by means of the substitutionx=y−n+^1.

Differential equations of the nth Order and of the nth degree

Consider special cases of linear differential equations.

Type I – The linear differential equation

(A)

dny

dxn

+P 1

dn−^1 y

dxn−^1

+P 2

dn−^2 y

dxn−^2

+···+Pny= 0

in which coefficientsP 1 ,P 2 ,...Pnare constants.
Consider the differential equation of third order

(B)

d^3 y

dx^3

+P 1

d^2 y

dx^2

+P 2

dy

dx

+P 3 y= 0

whereP 1 ,P 2 andP 3 are constants. The corresponding auxiliary equation is

r^3 +P 1 r^2 +P 2 r+P 3 = 0

Let the roots ber 1 ,r 2 ,r 3.
Ifr 1 ,r 2 ,r 3 are real and distinct,

y=C 1 er^1 x+C 2 er^2 x+C 3 er^3 x

Ifr 1 ,r 2 ,r 3 are real and equal

y=C 1 e−r^1 x+C 2 xe−r^2 x+C 3 x^2 e−r^3 x

In casea+bianda−biare each multiple roots of the auxiliary equation occur-
ringstimes, the solutions would be

C 1 eaxcosbx,C 2 xeaxcosbx,C 3 x^2 eaxcosbx,...Csxs−^1 eaxcosbx

C 1 ′eaxsinbx,C 2 ′xeaxsinbx,C 3 ′x^2 eaxsinbx,...Cs′xs−^1 eaxsinbx

Summary for the rule for solving differential equations of the type

dny

dxn

+P 1

dn−^1 y

dxn−^1

+P 2

dn−^2 y

dxn−^2

+···+Pny= 0

whereP 1 ,P 2 ,...Pnare constants.