Mathematical Methods for Physics and Engineering : A Comprehensive Guide

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

(in this example equal to 3). The RHS of a homogeneous ODE can be written as

a function ofy/x. The equation may then be solved by making the substitution

y=vx,sothat

dy

dx

=v+x

dv

dx

=F(v).

This is now a separable equation and can be integrated directly to give

∫

dv

F(v)−v

=

∫

dx

x

. (14.19)

Solve

dy

dx

=

y

x

+tan

(y

x

)

.

Substitutingy=vxwe obtain

v+x

dv

dx

=v+tanv.

Cancellingvon both sides, rearranging and integrating gives

∫

cotvdv=

∫

dx

x

=lnx+c 1.

But
∫
cotvdv=

∫

cosv

sinv

dv=ln(sinv)+c 2 ,

so the solution to the ODE isy=xsin−^1 Ax,whereAis a constant.

Solution method.Check to see whether the equation is homogeneous. If so, make

the substitutiony=vx, separate variables as in (14.19) and then integrate directly.

Finally replacevbyy/xto obtain the solution.

14.2.6 Isobaric equations

An isobaric ODE is a generalisation of the homogeneous ODE discussed in the

previous section, and is of the form

dy

dx

=

A(x, y)

B(x, y)

, (14.20)

where the equation is dimensionally consistent ifyanddyare each given a weight

mrelative toxanddx, i.e. if the substitutiony=vxmmakes it separable.