# Thread: Differential equation and homogeneous equation of degree n

1. ## Differential equation and homogeneous equation of degree n

A function f of two variables is said to be homogeneous of degree n if

f(tx,ty)=tnf(x,y)

whenever t>0.

Show that such a function f satisfies the equation

x(df/dx)=y(df/dy)=nf.

**('d's in above equation supposed to be curly d, but I couldn't figure out how to insert them)**

2. ## Re: Differential equation and homogeneous equation of degree n

What you have stated here is not true! $\displaystyle f(x, y)= x^2+ y^2$ is homogeneous of order 2: $\displaystyle f(tx, ty)= (tx)^2+ (ty)^2= t^2x^2+ t^2y^2= t^2(x^2+ y^2)= t^2f(x, y)$. But $\displaystyle x\frac{\partial f}{\partial x}= 2x^2$ and $\displaystyle y\frac{\partial f}{\partial y}= 2y^2$. They are not equal and not equal to 2f.

Did you mean $\displaystyle x\frac{\partial f}{\partial x}+ y\frac{\partial f}{\partial y}= n f$?

3. ## Re: Differential equation and homogeneous equation of degree n

Sorry, that's exactly what I meant - the '=' was a typo.