Estimate the value of $y_1(3)-y_2(3)$ with $y'=\arctan(y+x+\cos(y)^2)$ and $y_1(0)=1, y_2(0)=2$ Show that $y_1$ and $y_2$ exist, determin the sign and estimate the value of $y_1(3)-y_2(3)$ with $y'=\arctan(y+x+\cos(y)^2)$ and $y_1(0)=1, y_2(0)=2$ I already solved the first question, they exist and are unique because $\arctan(y+x+\cos(y)^2)$ is continuous and its partial derivatives are also continuous. How can i proceed? I don't think that determining an explicit solution is a good path also because i wouldn't know how to do it in this case.

Solutions are unique, so that the graphs of $y_1$ and $y_2$ do not cross. Since $y_1(0)