Register to reply

An ODE I was thinking of.

by MathematicalPhysicist
Tags: thinking
Share this thread:
MathematicalPhysicist
#1
Feb14-13, 10:26 AM
P: 3,243
I thought today of the next DE:

[tex] y''(x) = y(x)e^{y'(x)}[/tex]

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.

[tex]y(x)=\sum_{n=0}^{\infty} a_n x^n [/tex]

[tex]y'(x)=\sum_{n=0}^{\infty} na_n x^{n-1}[/tex]

[tex]y''(x)=\sum_{n=0}^{\infty} n(n-1)a_n x^{n-2} [/tex]

equating:

[tex] \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}} [/tex]

[tex]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 [/tex]

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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
JJacquelin
#2
Feb14-13, 11:14 AM
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
ODE Lambert.JPG  
MathematicalPhysicist
#3
Feb14-13, 11:37 AM
P: 3,243
Thanks.

MathematicalPhysicist
#4
Feb15-13, 08:50 AM
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:

[tex]y^{(n)}+(y^{(n-1)})^2+(y^{(n-2)})^3+\ldots + (y')^{n+1}+y^{n+2} = 0[/tex]

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
#5
Feb15-13, 09:02 AM
P: 3,243
Maybe some examples if the general case isn't clear enough.

for n=1:

[tex] y'+y^2=0[/tex]

For n=2:

[tex] y''+(y')^2+y^3=0[/tex]

For n=3:

[tex] y'''+(y'')^2+(y')^3+y^4=0[/tex]

Etc.


Register to reply

Related Discussions
Linear thinking Vs. Picture thinking Medical Sciences 7
Doing some thinking. Social Sciences 18
What is thinking? Medical Sciences 7
What is he/she thinking General Discussion 5
I was just thinking... General Physics 4