(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve [itex]y' = y^2 - xy +1 \qquad(1) \qquad ,[/itex] using the substitution y = x + 1/u

3. The attempt at a solution

Upon substitution, I arrive at

[tex] du/dx - xu = 1[/tex]

which is linear/1st order/non-homogenous. When I apply the integrating factor method, I arrive at

[tex]u(x) = e^{\frac{1}{2}x^2}\left ( \int e^{-\frac{1}{2}x^2}dx + C\right ) \qquad(2)[/tex]

I am not sure how to deal with (2) as it is non integrable in terms of elementary functions. When I solve (1) using Wolfram Alpha, the solution is in terms of the error function. We have not covered the error function in class since it is assumed to be prior knowledge. However, I have never used it in any classes, so I am unsure how to manipulate (2) using erf().

From wiki, I have the definition to be

[tex]erf(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt\qquad(3)[/tex]

and this is to be used to replace my integral in (2), namely

[tex]\int e^{-\frac{1}{2}x^2}dx + C\qquad(4)[/tex]

Now two things are confusing me here:

1) How to deal with the bounds in the integral in the definition of erf(x) since I am dealing with an indefinite integral in (4)?

2) How to deal with the factor of 1/2 that I have in (4)?

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# Homework Help: Ricatti's Equation (non linear ODE)

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