Recent content by DrWahoo

Using an integrating factor helps make the differential equations "Exact". Wonderful way to solve simple and some decently complex ode's if you can integrate the integrating factor. Can you check that once you multiply by the integrating factor that you do indeed have an exact ode?

I have to say I am guilty of this. Have been doing a better Job, but typically I only bring up trivial for the 0 case. We are discussing tangent bundles, currently, and getting into a lot of Li Algebra and topological spaces that they are not use to seeing in an ODE class.

Yes. Check out Wolfgang Walter's book on DE's. Or Roseenwasser's books on autonomous control systems using sensitivity analysis. Notation is similar in both.

David C. Lay's book Linear Algebra with applications would give some not so "trivial answers" to a "somewhat trivial question." Wonderful book for upper level undergrads and graduate students. Definitely worth a read. See attached for some samples.

Hint for #3. Convert to polar coordinates. Set the RHS of each DE to 0 and solve. What possible equilibrium points do we have? Can you follow the steps from #2 to get the answer? Another method is to think of solutions to a similar DE, instead of to the 5th power, what if the largest...

Gould's Home Page Vol 1 - 8 all contain good info on this type of stuff.

After you square both sides, make sure to multiply by $r$ to get rid of the fraction. Once you do that, you should have a simple equation to solve using multiplication and division. What are you getting as a solution without plugging in numbers for each unknown?

Could always use generating functions for the binomial.

These are problems my 600 level modeling students were given; I will give some hints and a complete proof and overview of problem two. Hopefully someone else can chime in and voice their ideas as I love seeing analytical thinking. On the 2nd problem we are asked; 2) Consider the following...

1) Reduce the quadratic system to its normal form; $x'=y+x^2 +y^2$ $y'=xy$ 2) Consider the following vector field on $R^2$ $x'=\mu x -y-x(x^2 +y^2)$ $y'=x+\mu y -y(x^2 +y^2) $ where $\mu\in\Bbb{R}$. Define the cross section $\Sigma$ to the vector field by $\Sigma=\{(r, \theta) \in \Bbb{R}$x$...

In the preable you have to dedicate packages to run. So yes, for this problem some simple packages from ams work. Most are included in the miktex and other repositories, however, some packages I have made are not on CTAN(mirrors) or in any package manage, so I simply include the "class" or...

Re: Integral - Oriantation Just a suggestion, you can always check using a gradient vector field to see what happens at that point. Check for the normal vector. What signs and which direction do they point?

Thanks mark. I will give a little intuition as well and let a student come up with the result. This is just a reiteration of Marks work. Based off of $\Bbb{R}$ pick to integrate with respect to $x$ first. This would imply; that the boundaries of the region in the $x$ direction would be...

Anyone want to attempt a solution or give some hints?

I can post a graph of the function, derivative, inverse, and solution in graph forum from MATLAB if that helps anyone. I must admit, the question is written in a very vague forum. It doesn't define everything how "most mathematicians" would. It leaves a lot of the problem open for...