How to solve \(\displaystyle{y}{''}+{\left({1}-{y}\right)}{\left({1}+{y}\right)}{y}={0}\)

Jackson Floyd

Jackson Floyd

Answered question

2022-03-30

How to solve
y+(1y)(1+y)y=0

Answer & Explanation

Cody Hart

Cody Hart

Beginner2022-03-31Added 11 answers

Step 1
y+(1y)(1+y)y=0
Swithch variables to get
x[x]3+(1y)(1+y)y=0
Reduction of order p=x gives
p=x=±1y42y2+c1
and now, you will face an ugly ellptic integral.
x+c2=±dyy42y2+c1
Rewrite
y42y2+c1
=12(y2a)(y2b)
where a and b are the roots of the quadratic in y2 and, hoping that I am not mistaken,
x+c2=±2bF(sin1(ya)ab)

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