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Assume that a continuous function $E(y)$ is such that $$E(0)=0,~~~~E(y)\neq 0,~~~~0<y\le 1.$$ Then $y=0$ is the singular solution of the differential equation $$ \dfrac{dy}{dx}=E(y), $$ if and only if the improper integral $$ \int_{0}^{1}\dfrac{dy}{E(y)}$$ is convergent.

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This problem is from china ODE problem (can see :http://item.jd.com/1048628556.html, page 111. problem 4, and I don't see any book have solution by this problem. Thank you

Lutz Lehmann
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math110
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    I can't understand the question. – Git Gud Apr 25 '14 at 12:37
  • Could you make a screenshot of the desired pages? (I don't know Chinese at all.) – Voliar Apr 25 '14 at 12:41
  • @GitGud,Now have you understand? – math110 Apr 25 '14 at 12:50
  • @math110 I still don't understand. You say "$y=0$ is differential equation $\dfrac{dy}{dx}=E(y)$". Do you mean to say that the constant function $\bf 0$ is a solution of $\dfrac{dy}{dx}=E(y)$? – Git Gud Apr 25 '14 at 13:11
  • I think I got it. Please try to understand if what's below is what you want to ask. $$\text{The constant function }\mathbf{0}\text{ is the only solution of }y'=E\circ y\text{ if, and only if, }\int \limits _0^1 \dfrac 1{E(y)}\mathrm dy\text{ converges}.$$ – Git Gud Apr 25 '14 at 13:14
  • yes,That's you mean,But my problem is signular solution – math110 Apr 25 '14 at 13:31

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