Some Homogeneous Ordinary Differential Equations

′

y

2-

y

x

′

u

+u2-u

x

′

u

2-2u

dx

x



du

2-2u

log(x)-log(C)-

1

2

log(2-2u)

x

C

2-2u

x

C

2-

2y

x

x

2

2-

2y

x

This Demonstration shows a procedure for solving an ordinary differential equation of the form

y'=f(y/x)

. The first step is to introduce a new variable

u=y/x

,

y=xu

. Differentiating the last equation, we get

y'=u+xu'

. By substitution, we get

xu'+u=f(u)

,

xu'=f(u)-u

. In the last equation, we separate variables to get

dx/x=du/(f(u)-u)

. Integration of both parts yields

log(x)-log(C)=∫1(f(u)-u)u

. From the last equation, we get a general solution of the form

x=Cg(u)

where

g(u)=

∫1(f(u)-u)u

e

.