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The differential inequality $$y^'\leq \frac{x^{2011}(1+y^6)}{y^2}$$ with $\displaystyle y\left(\frac{\pi}{4}\right)=2012$ is given? What can one say about $y(x)?$

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up vote 2 down vote accepted

You can translate the inequality to $\frac{y^2 \cdot y'}{1+y^6} \leq x^{2011}$ and then integrate from $x_00$ to $s$ with respect to $x$.

$$ \int_{x_0}^s \frac{y(x)^2 \cdot y(x)'}{1+y(x)^6} dx \leq \frac{s^{2012}}{2012}-\frac{x_0^{2012}}{2012}$$

$$ \frac{1}{3} \arctan y^3(s) -\frac{1}{3}\arctan y^3(x_0) \leq \frac{s^{2012}}{2012}-\frac{x_0^{2012}}{2012}$$

In your case, take $x_0=\frac{\pi}{4}$

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What you basically have is this:

\begin{align*} \frac{dy}{dx} & \leq \frac{x^{2011}\cdot (1+y^{6})}{y^{2}} \\ \Longrightarrow \int\frac{y^{2}}{1+y^{6}}\ dy &\leq \int x^{2011} \ dx \end{align*}

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