Can anyone offer an explanation for the powers (6 and 12) used in the Lennard Jones potential? (other than simply saying weak attractive and strong repulsion forces)
I am especially looking for short derivation that can be used to argue that the attractive potential falls off with the sixth...
It is a problem to take the exponential of a non-square matrix.
How can you calculate A^n when you can't multiply a non-square matrix with itself, its non conformable.
Nonlinear equations are often solved by making a substitution that puts the equation in a linear form. For the first problem, try multiplying through by y and making the substitution v=y^2 to yield
x v'(x) - 2v(x) = 8x^3
then solve, and substitute y back in. Looks like the second problem...
I got a slightly different answer than what you posted from the text
y(x) = -\sqrt{\frac{1}{2}(x^2+1)}
and mathematica agrees with me, so perhaps a typo?
Anyway, it looks like your on the right track, although go back through the integration, I think you may be off by a factor.
Then...
I am curious what the nonlinear bernoulli equation is used to model. Is there a certain topic or context where it shows up often? Can any suggest some references for more info?
I am reviewing some ODE stuff for an upcoming exam and would really like an intuitive feel for the equation and its...