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I have to solve $ y'(x)=-2xy(x)+ey^2(x) $.

Using $ z=y^{-1}$ and $-z^{'}=\frac{y^{'}}{y^{2}}$ i arrive to prove that $ z^{'}=-2xz+e $, but when i apply the variation of constants method i obtain $ z_0(x)=Ce^{A(x)}, A(x)=x^2\Rightarrow Ce^{x^2}$ and, unfortunately:

$ z_p(x)=e^{A(x)}B(x), B(x)=\int -e\cdot e^{-A(x)}=\int-e \cdot e^{-x^2}=\int-e^{1-x^2}$

How must i behave now? How can i arrive to the solution with Riemann's integral? How to use definite integrals to solve the EDO?

Thanks for any help!

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  • $\begingroup$ Use mathematica DSolve method: DSolve[y'[x] == -2 xy[x] + Ey[x]^2, y[x], x] $\endgroup$ – William Sep 24 '18 at 16:53
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$$y'(x)=-2xy(x)+ey^2(x)$$ Substitute $z=\frac 1 y$ $$-z'=-2xz+e \implies z'-2xz=-e$$ Use integrating factor $\mu=e^{-x^2}$ $$(ze^{-x^2})'=-ee^{-x^2}$$ Integrate $$ze^{-x^2}=-e\int e^{-x^2}dx+C$$ $$ze^{-x^2}=-e \frac {\sqrt {\pi}}2\text{erf(x)}+C$$ $$\frac y {e^{-x^2}}=\frac 1 {-e\frac {\sqrt {\pi}}2\text{erf(x)}+C}$$ $$ y =\frac {2e^{-(x^2+1)}} {C- {\sqrt {\pi}}\text{erf(x)}}$$

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    $\begingroup$ Thanks Isham for your answer. So, using the error function i obtain $ y=\frac{1}{e^{x^{2}}(c-e\frac{\sqrt{\pi }}{2}erf(x)))} $. However, my question was: how can i use definite integrals to solve the integral? WolframAlpha says that the solution is $ y(x)=\frac{2}{e^{c+x^{2}}+e}$, so i think it's possible to solve it without error function. $\endgroup$ – Marco Pittella Sep 24 '18 at 17:31
  • $\begingroup$ @MarcoPittella Are you sure for WA answer ? I don't think you can solve this differential equation without error function $\endgroup$ – Isham Sep 24 '18 at 17:36
  • $\begingroup$ y're welcome @MarcoPittella $\endgroup$ – Isham Sep 24 '18 at 17:47
  • $\begingroup$ You're right: i've added an $x$ to the text. Thanks again! $\endgroup$ – Marco Pittella Sep 24 '18 at 17:47
  • $\begingroup$ Do you think it's possible to solve it with definit integrals? $\endgroup$ – Marco Pittella Sep 24 '18 at 17:48

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