Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $y(x)?$

share|improve this question
add comment

2 Answers

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 $x_0=\frac{\pi}{4}$

share|improve this answer
add comment

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*}

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.