A Solution form for the following differential equation

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!!
 

RPinPA

Science Advisor
Homework Helper
506
283
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]
 
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
---
 

Want to reply to this thread?

"Solution form for the following differential equation" You must log in or register to reply here.

Related Threads for: Solution form for the following differential equation

Replies
5
Views
2K
Replies
3
Views
2K
Replies
10
Views
826
Replies
20
Views
3K
Replies
2
Views
1K
Replies
20
Views
2K
Replies
6
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top