How to solve this equation by integration \(\displaystyle{\frac{{{x}’’}}{{{x}}}}-{\frac{{{x}’}}{{{x}^{{2}}}}}-{\frac{{{x}’^{{2}}}}{{{x}^{{2}}}}}={0},\)

Landin Harmon

Landin Harmon

Answered question

2022-03-30

How to solve this equation by integration
xxxx2x2x2=0,

Answer & Explanation

Ronald Martinez

Ronald Martinez

Beginner2022-03-31Added 7 answers

Step 1
xxxx2x2x2=0,
×x2x2=xx2
(xx)=(1x)
Reduce the order of the DE by integration.
Step 2
Edit
(xx)=(1x)
xx=1x+C
x=Cx1
This DE is separable.
dxCx1=dy
delai59qk

delai59qk

Beginner2022-04-01Added 8 answers

Step 1
There is another way to solve this equation : switch variables and write it as
1xy[y]31x21y1x21[y]2=0
that is to say
xy+y+[y]2=0
and the reduction of order becomes obvious. With p=y
xp+p+p2=0
p=1q
xq
=1+qq
=c1x1
p
=1c1x1
Just finish and inverse.

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