Can This Nonlinear ODE with Exponential Terms Be Solved Using Power Series?

[MathematicalPhysicist]

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.

 

[JJacquelin]

Hi !
The ODE is solvable on the form of the inverse function x(y) as a special function defined by an integral :

 

[MathematicalPhysicist]

Thanks.

 

[MathematicalPhysicist]

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

 

[MathematicalPhysicist]

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.