Solving Homogeneous Differential Equations
\[f(x,y)\]
is homogeneous if \[f(tx,ty_=t^nf(x,y)\]
for some \[n\]
. If \[n=0\]
then we can write \[f(x,y)\]
as a function of one argument \[f(1,y/x=v)\]
. We can use this to solve homogeneous differential equations.Example
\[\frac{dy}{dx}= \frac{y^3-2x^3}{xy^2}\]
.\[ \frac{(ty)^3-2(tx)^3}{(tx)(ty)^2}= \frac{y^3-2x^3}{xy^2}\]
.Let
\[y=vx\]
then \[\frac{dy}{dx}=v+ x \frac{dv}{dx}\]
.The equation becomes
\[v+ \frac{dv}{dx} = \frac{(vx)^3-2x^3}{x(vx)^2}=\frac{v^3-2}{v^2} \rightarrow x \frac{dv}{dx}=-\frac{2}{v^2}\]
.Separating variables gives
\[v^2 \frac{dv}= - \frac{2}{x} dx \rightarrow \int v^2 \frac{dv}=- 2 \int \frac{2}{x} dx \rightarrow \frac{v^3}{3}=-2 lnx+c\]
.Using
\[y=vx \rightarrow v= y/x\]
gives \[\frac{y^3}{3x^3}=-2lnx+c \rightarrow y^3=3x^3(c-2lnx)\]
. 
