I'm doing the following problem: The differential equation $$ \dot{y} = X(t,y), X(t,y) = \frac{1}{3}y^{1/4} +t^{1/3}$$ defined on $D_X = (0,\infty)\times(0,\infty)$.
I already solved it with: $y(t) = t^{4/3}$.
But here is what i don't understand. The problem says:
For $\eta >0$ let $(I_\eta,y_\eta)$ denote the maximal solution with $y_\eta(1) = \eta$
for:
a) For $0 < \eta < 1 $ Then $y_\eta(t) < t^{4/3}$, for $t \in I_\eta$
b) For $\eta > 1 $ Then $y_\eta(t) > t^{4/3}$, for $t \in I_\eta$
I am very confused about the $y_\eta(1) = \eta$ notation, so i can't understand what the goal with the task is. Can you help?