Differential equation notation about maximal solution

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?

• $y_\eta$ denotes the maximal solution of the equation satisfying the initial condition $y(1)=\eta$. $I_\eta$ stands for the domain of that maximal solution. You have obtained a solution of the equation for $\eta=1$ only. – user539887 Apr 5 at 11:20
• I see. Thank you – Pernk Dernets Apr 8 at 6:12