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I tried this with Euler equation but I end up with a very hard formula to integrate...I did several similar problem but keep on getting stuck at the final integration. Please help.
Hey Tennis, I believe we can obtain an implicit solution. Some background for those not familiar with the problem: We wish to minimize the following integral over some domain D:
. Euler's necessary requirement to do this is the following:
Plugging all that in and remembering the partials treat y and y' as just regular variables but the ordinary derivativetreats them as functions of x, I get (if I didn't make any errors) the following ODE:
Ok, that's a little intimidating but we can do the first integration by making the change of variableand integrating. I get:
Now integrate the inverse ofat the point
to obtain an implicit albeit not very satisfying expression for the solution:
} H^{-1}(c_1-\frac{1}{2y})dy=\int_{x_0}^{x} dx)
I've made some additional progress with this which I find really interesting and hope some of you will too. Ignoring for the moment extremizing the integral, I'll focus on just the IVP:
LettingI obtain:
or:
This gives:
Now note that
Therefore if I differentiate y with respect to p, divide by p, I'll have dx or:
}-\frac{\arctan(p)}{2}+c_1\right)^{-1}\right\})
and therefore:
Integrating:
I now haveand
as the parametric solution to the DE in terms of the variable
which is still the derivative of the function. So I've expressed the solution parametrically as a function of it's derivative! That's pretty cool I think.
However, being the skeptical person I am about all this, I checked it numerically. Solving for
above, I solved the DE numerically and then superimposed the results against the parametric solution (x(p) calculated numerically). The numerical solution is red, parametric solution is blue. Pretty close.
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