I am learning differential equation and the uniqueness theorem for 1st order ODE said that if $y' = f(x,y)$ where $f(x,y)$ is Lipschitz w.r.t. $y$ or $\frac{df}{dy}$ is continuous, then the ODE has at most one solution. But my question is not about this theorem.
Now, I find this homogeneous ODE: $$ y' = \frac{xy + y^2 + x^2}{x^2}$$ The usual technique applies and a solution is found. My question is that
Is $\frac{xy + y^2 + x^2}{x^2}$ Lipschitz?
Do we have another solution for this ODE so that the contrapositive of the uniqueness theorem is verified?
I tried this $$ \frac{f(x,y)-f(x,0)}{x-0}=\frac{xy+y^2}{x^3}$$ Can I argue that when $y$ tends to infinity, the above diverges?
Thanks in advance.
$$ \frac{\partial f}{\partial y}=\frac{x+2\,y}{x^2}. $$ It is continuous on $\{(x,y):x>0\}$, and hence $f(x,y)$ is (locally) Lipschitz with respect to $y$ on the same set. The existence and uniqueness theorem tells you that given $x_0>0$ and $y_0\in\mathbb{R}$ there is a unique solution defined on some interval around $x_0$ such that $y(x_0)=y_0$. To find it, you impose that condition on the general solution to find the value of the integration constant $C$.