- #1
badtwistoffate
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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?
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?