Solutions of a first-order ODE

Question

Given a differential equation $y' = f(x,y)$, where $f$ is continuous. Suppose that $y_1, y_2$ are two solutions of the equation on the point $(x_0, y_0)$. Moreover it is known that $y_1$ and $y_2$ are not identical in $(x_0, x_1) \;\forall x_1>x_0$ and $y_1 \geq y_2$. Is it possible for $y_1$ and $y_2$ to have equal value in a discrete set of points, converging to $x_0$ from right?

Answer 1

Consider $y_2(x) = x$ and $y_1(x) = x + x^4 \sin(1/x)^2$, $x_0 = 0$, $y_0 = 0$. For these to be solutions of the differential equation, we need $f(x,x)=1$ and $f(x,x+x^4 \sin(1/x)^2) = 1 + 4 x^3 \sin(1/x)^2 - 2 x^2 \sin(1/x) \cos(1/x)$. For example, you could have $$ f(x,y) = \cases{1 + 2 (y-x)^{1/3} x^{2/3} |\sin(1/x)|^{1/3} \left(2 x \sin(1/x) - \cos(1/x)\right) & for $y \ge x \ge 0$\cr 1 & otherwise}$$