What is the issue with the two methods used to solve the ODE dy/dx=x(1-y)?

[badtwistoffate]

Got the eqn dy/dx=x(1-y) and it can be solved both linear and separable methods.(Linear method being using a integrating factor) Problem I am having is that with this two methods i get two different (yet similar answers) and was wondering if you can see my problem with this two methods I am using.

Integrating Factor method:
y'+xy=x, u'(x)=e^(x^2/2)

[e^(x^2/2)y]'=x*e^(x^2/2)

e^(x^2/2)y=integral(x*e^(x^2/2)), do u substitution, get...

e^(x^2/2)y=e^(x^2/2)+c

y=1+c/e^(x^2/2) or y=1+c*e^(-x^2/2)

Separable method:
dy/(1-y)=x dx, integrate both sides

-ln(1-y)=e^(x^2/2)+C, raise both sides to e.

1/(1-y)=K*e^(x^2/2)+C, rearrange to get y=.

y=1-1/K*e^(x^2/2)

so we get two different answers with these methods, where is the problem lieing or are both wrong?

 

[Galileo]

Separable method:
dy/(1-y)=x dx, integrate both sides

-ln(1-y)=e^(x^2/2)+C, raise both sides to e.

1/(1-y)=K*e^(x^2/2)+C, rearrange to get y=.

y=1-1/K*e^(x^2/2)

You got a bit sloppy near the end. Some mistakes are just typo's I think.

$$-\ln(1-y)=\frac{1}{2}x^2+C$$
$$1-y=K\exp(-\frac{1}{2}x^2)$$
$$y=1-K\exp(-\frac{1}{2}x^2)$$

So it's the same.