# An ODE I was thinking of.

by MathematicalPhysicist
Tags: thinking
 P: 3,243 I thought today of the next DE: $$y''(x) = y(x)e^{y'(x)}$$ Not sure if it has applications, obviosuly I tried to find a solution via power series around x=0. It seems tough to look for a general recurrence equation for the coefficients. Here's what I have done so far. $$y(x)=\sum_{n=0}^{\infty} a_n x^n$$ $$y'(x)=\sum_{n=0}^{\infty} na_n x^{n-1}$$ $$y''(x)=\sum_{n=0}^{\infty} n(n-1)a_n x^{n-2}$$ equating: $$\sum_{n=0}^{\infty} n(n-1)a_n x^{n-2} =\sum_{n=0}^{\infty} a_n x^n e^{\sum_{n=0}^{\infty} a_n n x^{n-1}}$$ $$e^{a_1} e^{2a_2 x} e^{3a_3 x^2} \cdots = e^{a_1}[1+2a_1 x + \frac{(2a_1 x)^2}{2!}+\cdots]\cdot [1+3a_3 x^2 +\frac{(3a_3 x^2)^2}{2!}+\cdots]\cdot \cdots$$ I am not sure if it even converges, is this equation known already, I am quite sure someone already thought of it. Thanks in advance.
 P: 759 Hi ! The ODE is solvable on the form of the inverse function x(y) as a special function defined by an integral : Attached Thumbnails
 P: 3,243 Thanks.
 P: 3,243 An ODE I was thinking of. Well, if I am already in the mood for non-ordinary DEs, I'll make this thread a thread with peculiar DEs I have in my mind. Here's another one: $$y^{(n)}+(y^{(n-1)})^2+(y^{(n-2)})^3+\ldots + (y')^{n+1}+y^{n+2} = 0$$ Guessing a solution in the form of power series will be hard work (which I don't have time for right now), so does it have an specail function form solution? P.S n\geq 1
 P: 3,243 Maybe some examples if the general case isn't clear enough. for n=1: $$y'+y^2=0$$ For n=2: $$y''+(y')^2+y^3=0$$ For n=3: $$y'''+(y'')^2+(y')^3+y^4=0$$ Etc.

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