expeditiupc

Answered

2022-01-21

Closed form solutions of $\ddot{x}\left(t\right)-x{\left(t\right)}^{n}=0$

Answer & Explanation

Paul Mitchell

Expert

2022-01-21Added 40 answers

After here you can change variable and use the binomial expansion to evaluate integral.

I avoided doing many calculations, and I preferred to use the quick way: ask Wolfram Alpha.

kaluitagf

Expert

2022-01-22Added 38 answers

This is the answer given by Mathematica: $x=x\left(t\right)$ is implicitly given by

$(n+1)x{\left(t\right)}^{2}{({c}_{1}n+{c}_{1}+2x{\left(t\right)}^{n+1})}^{2}$

$\frac{2{F}_{1}{(\frac{1}{2},\frac{1}{n+1};1+\frac{1}{n+1};-\frac{2x{\left(t\right)}^{n+1}}{n{c}_{1}+{c}_{1}})}^{2}}{({c}_{1}n+{c}_{1}){({c}_{1}(n+1)+2x{\left(t\right)}^{n+1})}^{2}}={({c}_{2}+t)}^{2}$ .

In general, I do not believe that y may be written down in terms of elementary or special functions.

In general, I do not believe that y may be written down in terms of elementary or special functions.

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