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Can one give a hint how to solve the following equation?


Thanks in advance.

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Would you be fine with an implicit solution? – J. M. Nov 5 '11 at 14:15
Nope, but I like wolframalpha too :) – Daria Morys Nov 5 '11 at 14:18
this could help – pedja Nov 5 '11 at 15:52
$y(x)=\sqrt{x}$ does not count, right? Would you be interested to know $x(y)$? – Aleksey Pichugin Nov 5 '11 at 15:53
up vote 2 down vote accepted

For this one we look for an integrating factor which will make the equation exact.

Let $M = x - 2xy - y^2$ and $N = y^2$. Then $M_x = 1-2y$ and $N_y = 2y$.

Thus $$\frac{M_x-N_y}{N} = \frac{1-2y}{y^2} = \frac{1}{y^2} -2 \frac{1}{y}$$ which is purely a function of $y$. This tells us that there is an integrating factor $\mu (y)$ which makes the equation exact which satisfies

$$ \frac{d\mu}{dy} = \left( \frac{1}{y^2} -2 \frac{1}{y} \right)\mu $$

Clearly $$\mu(y) = e^{\displaystyle\int\left( \frac{1}{y^2} -2 \frac{1}{y} \right)dy} = e^{\frac{-1}{y}-2\log y} = y^{-2} e^{y^{-1}}.$$

Try (first checking that my algebra is correct) and then multiplying the differential equation by this integrating factor, it should become an exact differential equation which I'm guessing you know how to solve if you are asking this question.

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Hint: Your equation can be turned into a first order linear differential equation in $\frac{dx}{dy}$ (i.e., just multiply the equation by $\frac{dx}{dy}$ and it is a standard form....)

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BTW, the solution is also a simple looking one, if you can't figure out, please let us know. – Tapu Nov 5 '11 at 15:34
What standard form might that be? $x-2xy-y^2$ isn't homogeneous, so dividing it by $y^2$ won't make it a function of $v=x/y$ if that's what you're thinking. – anon Nov 5 '11 at 16:03
@anon, No, a linear 1st order ODE has the standard form $\frac{dx}{dy}+P(y).x=Q(y)$ and for it $e^{\int P(y) dy}$ is always an integrating factor. Yes, it was that much simple!! – Tapu Nov 5 '11 at 16:33
Oh, of course. You're right. – anon Nov 5 '11 at 23:56

As it looks some people misunderstood (I apologize, the fault may be my side) my first answer, here is the complete solution after reducing the problem into standard form:

As I wrote in the first answer, multiplying both sides by $\frac{dx}{dy}$ (the hint was that it will make the equation linear in $x$),

$\frac{dx}{dy}+\left(\frac{1-2y}{y^2}\right).x=1$. This is a linear equation (in $x$) of the standard form $ \frac{dx}{dy}+P(y).x=Q(x)$. So, $e^{\int P(y)dy}=e^{\int \left(\frac{1-2y}{y^2}\right)dy}=\frac{e^{\frac{-1}{y}}}{y^2}$ is an integrating factor (IF). So, multiplying by the IF and integrating the general solution is given by


Or, $x=y^2(1+ce^{\frac{1}{y}})$.

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Please check the algebra, I do silly mistake very often. – Tapu Nov 5 '11 at 17:00

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