# ordinary differential equations – Initial value problems for \$y’=frac{x+y}{x-y}\$

The solutions of the differential equation $$y’=frac{x+y}{x-y}$$, are given implicitly by the relation $$ln x = arctanfrac{y}{x}-frac{1}{2}ln(1+frac{y^2}{x^2})+c,enspace cinBbb{R}.$$

I’m considering the existence and uniqueness of arbitrary initial value problem $$y(x_0) = y_0$$. Let’s say we write the equation as $$f(x,y)=0$$. The function will be continuously differentiable in $$Bbb{R^2}$$. Then by the implicit mapping theorem, if we have points $$ain A$$, $$bin B$$ such that $$f(a,b)=0$$ where $$A,B$$ are open sets. Then for each $$x_0in A$$ there will be unique solution $$y(x_0)in B$$, which is differentiable and therefore continuous. Am I missing some key insights here?