Closed form solutions of x¨(t)−x(t)n=0

expeditiupc

expeditiupc

Answered

2022-01-21

Closed form solutions of x¨(t)x(t)n=0

Answer & Explanation

Paul Mitchell

Paul Mitchell

Expert

2022-01-21Added 40 answers

yyn=0
yy=yyn
yydx=yyndx
(y)22=yn+1n+1+c1
y=2yn+1n+1+2c1
dy2yn+1n+1+2c1=dx=x+a
1(2c1)12dy1+yn+1c1(n+1)=x+a
dy1+yn+1c1(n+1)=(2c1)12x+a(2c1)12=(2c1)12x+c2
After here you can change variable and use the binomial expansion to evaluate integral.
u=yn+1c1(n+1)
(1+u)α=n=0(αn)un=1+αu1!+α(α1)u22!+...
I avoided doing many calculations, and I preferred to use the quick way: ask Wolfram Alpha.
k=1c1(n+1)

kaluitagf

kaluitagf

Expert

2022-01-22Added 38 answers

This is the answer given by Mathematica: x=x(t) is implicitly given by
(n+1)x(t)2(c1n+c1+2x(t)n+1)2
2F1(12,1n+1;1+1n+1;2x(t)n+1nc1+c1)2(c1n+c1)(c1(n+1)+2x(t)n+1)2=(c2+t)2.
In general, I do not believe that y may be written down in terms of elementary or special functions.

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