# Solution form for the following differential equation

Josu Aguirrebeitia
Hi. After arranging the dynamic contact between a elastic ball against a flat, I have reached the following differential equation for the motion during the contact:

m·x’’+(k+c·x’)·x^n=0

with m,c,k>0 and for exponent n --> 1<n<2

Any functional form for this equation? I have solved it numerically but I would love if any functional analytical form exist...

Thanks!

Homework Helper
I tried sending it to Wolfram Alpha, which I think is run on Mathematica. It required the Pro package (which I have) to solve it, but even so what I got was a complicated integral expression involving the Lambert W function rather than a closed form analytical expression.

https://www.wolframalpha.com/input/?i=m·x(t)’’+(k+c·x(t)’)·x(t)^n=0

Solution: $$\alpha_1 + t = \int_1^{x(t)} \frac {c} {k\left( -W\left(\frac {-\sqrt[n+1]{\exp\left(\frac{c^2 \xi^{n+1}}{km}\right) - n - \frac{c^2n\alpha_1}{k} - \frac{c^2\alpha_1}{k} - 1}} {k}\right) \right) - k} d\xi$$
But then it also said the computation time was exceeded, so maybe there was more it could do with this.

If you have access to Mathematica, this is the code it generated to evaluate the expression.
Code:
DSolve[{x[t]^n (k + c x'[t]) + m x''[t] == 0}, x[t], t]

Josu Aguirrebeitia
Josu Aguirrebeitia
Thank you for your time RPinPA, I appreciate. However, a numerical procedure is to be used in this case too, I solved the equation numerically for t with a explicit scheme, but I am after a closed analytical form of the solution... maybe impossible...

Thank you again!

J
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