Im solving initial value problem $ \frac {dy}{dx} + xy = xy^2; y(0)=3$ After applying Bernoulli's equation method i obtained $ \frac {dv}{dx} -xv = -x$ So, $ p(x) = -x, q(x) = -x $ For finding integrating factor $u(x)=e^ {-\int xdx}=e^ {-\frac {x^2}{2}}$ $ y= \frac {\int u(x)q(x)dx+C}{u(x)}={\int u(x)q(x)dx}$ So, $-\int xe^ {-\frac {x^2}{2}}dx $ Please help further or guide me if i did something wrong.
Solving initial value problem
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$\begingroup$
calculus
ordinary-differential-equations
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0Thanks i didn't notice that :) – 2012-11-23
3 Answers
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This seems fine. the only point is $v=\frac1y$, so $y=\frac1v$ and $u(x)v(x)-1\cdot\frac13=u(x)v(x)-u(0)v(0)=\int_0^x (u(t)v(t))'=\int_0^x u(t)q(t)dt=-\int_0^x te^{-\frac{t^2}{2}}dt$ So $v(x)=e^{\frac{x^2}{2}}\left(\frac13-\int_0^x te^{-\frac{t^2}{2}}dt\right),\hspace{10pt} y(x)=\frac{1}{v(x)}$
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1Since $v=\frac1y$, we have $v(0)=\frac13$ and I used it at the very beginning of the first line. – 2012-11-23
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Why don't you try separation of variables? This leads to a closed form solution that exists for all $x < \sqrt{2 \log \frac{3}{2}}$.
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0there is an error in the original post, see my comments. – 2012-11-23
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$ \mbox{Set}\ {\rm y}\left(x\right) \equiv {\rm f}\left(x^{2}\right)\equiv{\rm F}\left(x\right) $
$ \mbox{You get}\quad {\rm F}'\left(x\right) ={{\rm F}^{2}\left(x\right) - {\rm F}\left(x\right)\over 2} $