# Finding region of xy plane for which differential equation has a unique solution

#### find_the_fun

##### Active member
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point $$\displaystyle (x_0, y_0)$$ in the region.

$$\displaystyle x \frac{dy}{dx} = y$$

What does an xy-plane have to do with anything? I looked up the definition of unique solutions and here it is

Let R be a rectangular region in the xy-planed defined by a <=x<=b, c<=y<=d that contains the point $$\displaystyle (x_0, y_0)$$ in its interior. If f(x,y) and $$\displaystyle \frac{\partial{d} f}{\partial{d} y}$$ are continuous on R then there exists some interval $$\displaystyle I_0: (x_0-h, x_0+h), h>0$$ contained in [a/b] and a unique function y(x) defined on $$\displaystyle I_0$$ that is a solution of the initial value problem.

That's a bit difficult to digest. How do I proceed?

#### Fernando Revilla

##### Well-known member
MHB Math Helper
In each of the regions $D\equiv x>0$ and $D'\equiv x<0$ the differential equation is equivalent to $y'=f(x,y)=\dfrac{y}{x},$ and in both regions, $f$ and $\dfrac{\partial f}{\partial y}=\dfrac{1}{x}$ are continuous, so and according to a well known theorem, $D$ and $D'$ are solutions to your question.