Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.3 GENERAL ORDINARY DIFFERENTIAL EQUATIONS

15.3.4 Isobaric or homogeneous equations

It is straightforward to generalise the discussion of first-order isobaric equations

given in subsection 14.2.6 to equations of general ordern.Annth-order isobaric

equation is one in which every term can be made dimensionally consistent upon

givingyanddyeach a weightm,andxanddxeach a weight 1. Then thenth

derivative ofywith respect tox, for example, would have dimensionsminy

and−ninx. In the special casem= 1, for which the equation is dimensionally

consistent, the equation is called homogeneous (not to be confused with linear

equations with a zero RHS). If an equation is isobaric or homogeneous then the

change in dependent variabley=vxm(y=vxin the homogeneous case) followed

by the change in independent variablex=etleadstoanequationinwhichthe

new independent variabletis absent except in the formd/dt.

Solve

x^3

d^2 y

dx^2

−(x^2 +xy)

dy

dx

+(y^2 +xy)=0. (15.84)

Assigningyanddythe weightm,andxanddxthe weight 1, the weights of the five terms
on the LHS of (15.84) are, from left to right:m+1,m+1, 2m,2m,m+1. For these
weights all to be equal we requirem= 1; thus (15.84) is a homogeneous equation. Since it
is homogeneous we now make the substitutiony=vx, which, after dividing the resulting
equation through byx^3 ,gives

x

d^2 v

dx^2

+(1−v)

dv

dx

=0. (15.85)

Now substitutingx=etinto (15.85) we obtain (after some working)

d^2 v

dt^2

−v

dv

dt

=0, (15.86)

which can be integrated directly to give

dv

dt

=^12 v^2 +c 1. (15.87)

Equation (15.87) is separable, and integrates to give

1

2 t+d^2 =

∫

dv

v^2 +d^21

=

1

d 1

tan−^1

(

v

d 1

)

.

Rearranging and usingx=etandy=vxwe finally obtain the solution to (15.84) as

y=d 1 xtan

( 1

2 d^1 lnx+d^1 d^2

)

.

Solution method.Assume thatyanddyhave weightm, andxanddxweight 1 ,

and write down the combined weights of each term in the ODE. If these weights can

be made equal by assuming a particular value formthen the equation is isobaric

(or homogeneous ifm=1). Making the substitutiony=vxmfollowed byx=et

leads to an equation in which the new independent variabletis absent except in the

formd/dt.