Consider the IVP: $$y'(x)=f(x,y(x))\\y(x_0)=y_0$$ It is known that:
- If $f$ is continuous and Lipschitz in 2nd variable then: Existence and Uniqueness
- If $f$ is continuous then: just Existence
Now consider the following example that fails the above conditions: $$y'(x)=-\frac{x}{y(x)}\\y(0)=0$$
Since any solution satisfies $y^2(x)=-x^2$ the problem does not admit any real solution. But, it has the two imaginary solutions $$y(x)=\pm ix$$
My question: Are there any conditions on $f$ that guarantee existence of a complex solution to the general problem?