Answer to Question #260112 in Differential Equations for Jah

Question

Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0)in the region.

(x^2+y^2)y prime=y^2

Answer 1

According to theorem of Existence of Unique Solution if $f(x,y)$ and $df/dy$ are continuous

on rectangular region $R$ then there is exist interval $I$ on which unique exists.

We have:

$f(x,y)=\frac {y^2}{x^2+y^2}$

$\frac {df}{dy}=\frac {2y(x^2+y^2)-y^2(x^2+2y)}{(x^2+y^2)^2}$

$f(x,y)$ and $df/dy$ are not continuous at (0,0)

So, a unique solution exists in the region consisting of all points in the xy-plane except (0,0)