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Some Homogeneous Ordinary Differential Equations

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general solution
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constant in particular solution
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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
.
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