$\frac{\partial x}{\partial t}+x^3=yx^2$, $x(0)=x_{0}$
,where $x$ : $\mathbb{R}^+ $-> $\mathbb{R} $ and $y$ : $\mathbb{R}^+ $-> $\mathbb{R} $ with $\int_{0}^{t} y^2(s) ds < \infty$ for all $t$
if $x(t)$ is continuous function satisfying the above equation in $[0,T]$
I need to prove below inequalities. In order to show these, do I need to solve above differential equation? and if so, how can I apply above conditions to prove following inequalities?
