homogeneous equation
$$f(x,y)=0,$$ 
where the left hand is a homogeneous polynomial^{} of degree $r$ in $x$ and $y$, determines the ratio $x/y$ between the indeterminates^{}. One can be persuaded of this by dividing both of the equation by ${y}^{r}$. Then the left depends only on $x/y$ (which may be denoted e.g. by $t$).
Examples

•
The equation $5x+8y=0$ determines that $x/y=\frac{8}{5}$.

•
The equation ${x}^{2}7xy+10{y}^{2}=0$ determines that $x/y=2$ or $x/y=5$ (these values are obtained by first dividing both of the equation by ${y}^{2}$ and then solving the equation ${(x/y)}^{2}7(x/y)+10=0$).

•
The equation $2{x}^{3}{x}^{2}y6x{y}^{2}+3{y}^{3}=0$ determines that $x/y=\frac{1}{2}$ or $x/y=\pm \sqrt{3}$ (first divide the equation by ${y}^{3}$ and then solve $2{(x/y)}^{3}{(x/y)}^{2}6(x/y)+3=0$).
Title  homogeneous equation 
Canonical name  HomogeneousEquation 
Date of creation  20130322 15:14:41 
Last modified on  20130322 15:14:41 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  7 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 26C05 
Classification  msc 26B35 
Classification  msc 00A99 
Related topic  Variation 
Related topic  HomogeneousPolynomial 
Related topic  Equation 
Related topic  RegularDecagonInscribedInCircle 