Solving for y: Where Did My Calculations Go Wrong?

[asdf1]

for the following question:
y`+2y=y^2

my problem:
suppose u=1/y
so u`=-[y^(-2)]*y`=-1+2u
so du/(1+2u)=-dx
=>1+2u=ce^(-x)
=>u=[1-ce^(-x)]/2
so y=1/u=2/[1-ce^(-x)]

but the correct answer should be y=x/[1+2cx^(2x)]
does anybody know where my calculations went wrong?

 

[Fermat]

for the following question:
y`+2y=y^2
my problem:
suppose u=1/y
so u`=-[y^(-2)]*y`=-1+2u
so du/(1+2u)=-dx
=>1+2u=ce^(-x)
=>u=[1-ce^(-x)]/2
so y=1/u=2/[1-ce^(-x)]
but the correct answer should be y=x/[1+2cx^(2x)]
does anybody know where my calculations went wrong?

missed out dividing by 2 when integrating.

so du/(1+2u)=-dx
(1/2)ln(1+2u) = -x +lnC
ln(1+2u) = -2x + lnC²
ln{(1+2u)/C²} = -2x
1+ 2u = C²e^(-2x)
=>1+2u=ce^(-2x)

 

[asdf1]

opps~ thanks! :)