Is this an homogeneous equation?

[iVenky]

Actually I can't find if a differential equation is homogeneous or not

I thought homogeneous is given by

dy/dx= f(x,y)/ g(x,y)

but it doesn't look like that


For eg:

dy/dx= (y+x-1)/(y-x+2) is not homogeneous at all though

f(x,y)=y+x-1 and g(x,y)=y-x+2

How can you tell that if an equation is homogeneous or not?

Thanks a lot :)

 

[Vargo]

Hello.

Well, your definition is not right. $y'=f(x,y)$ is homogeneous if $f(tx,ty)=f(x,y)$ for any nonzero t. You can use that as a test. Basically, if you substitute tx and ty for x and y, then the t should divide itself out.

Example: $y'=\frac{x^2+xy}{\sqrt{x^4+y^3x}}$.
$\frac{(tx)^2+(tx)(ty)}{\sqrt{(tx)^4+(ty)^3(tx)}}= \frac{t^2(x^2+xy)}{\sqrt{t^4(x^4+y^3x)}}=\frac{x^2+xy}{\sqrt{x^4+y^3x}}.$