Consider $$ y'_1(t)=f_1(t),\qquad y_1(0)=y_{10}$$ and $$ y'_2(t)=f_2(t),\qquad y_2(0)=y_{20}. $$
If $f_1>f_2,\quad y_{10}>y_{20}$, then $$y_1>y_2.$$
The above is what I was told by my advisor, could anyone let me know where in what textbook I can find a standard theorem to back up this claim? (I looked through several ODE texts but not able to find the right theorem.)
Just integrate \begin{align*} y_{k}\left(t\right) & =y_{k0}+\int_{0}^{t}f_{k}\left(s\right)ds \end{align*} and compare the two forms. You are assuming, implicitly, that the $f_{k}$s are nice. Then, $$y_{1}\left(t\right)=y_{10}+\int_{0}^{t}f_{1}\left(t\right)dt>y_{20}+\int_{0}^{t}f_{2}\left(t\right)dt=y_{2}\left(t\right).$$