Some sunny day

The differential equation becomes $\frac{d}{dx}\big(e^{\frac12x^2}u\big) \; = \; \frac{x}{1+x}e^{\frac12x^2}$ and hence the general solution is $y_A(x) \; = \; Ae^{-\frac12x^2} + e^{-\frac12x^2}\int_0^{x^2}\frac{e^u}{1+u}\,du$ for some constant $A$ . If $y_A(x_0) = y_B(x_0)$ , then $e^{-\frac12x_0^2}A \; = \; e^{-\frac12x_0^2}B$ and hence $A = B$ , so $y_A \equiv y_B$ . Thus two distinct solutions of this differential equation cannot meet at a point.

The differential operator satisfies the uniqueness requirement of Picard's theorem for all initial conditions.