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Today I gave students the ODE exam. Originally, I want them to solve this nonlinear ODE: $$yy''=(y')^2,$$ which can be solved by using the substitution $u=y'$ and $y''=\displaystyle\frac{d}{x}u=\frac{du}{dy}\cdot\frac{dy}{dx}=u\frac{dy}{du}$. However, I missed the square and it became $$yy''=y'.$$ Since I didn't cover the power series method in the class, I wonder if it can solved it without using power series method. It seems to me that it doesn't work using the same substitution $u=y'$.

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The solution is an implicit equation involving the exponential integral $\int\frac{\exp\,x}{x}\mathrm dx$... –  J. M. Dec 14 '11 at 3:27
    
So there is no explicit form of $y$? –  Paul Dec 14 '11 at 3:56
    
Not as far as I know. –  J. M. Dec 14 '11 at 3:58
    
oh no...my poor students, they may spend a lot of time finding the explicit form of $y$. –  Paul Dec 14 '11 at 4:03
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Then may I suggest giving it away as a bonus for the trouble they've had? –  J. M. Dec 14 '11 at 4:04

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