# Nonlinear second order differential equation

• Dec 22nd 2012, 08:00 AM
JulieK
Nonlinear second order differential equation
What is the solution of the follwoing differential equation

$\frac{\partial^{2}y}{\partial x^{2}}-ay^{-1}\frac{dy}{dx}=0$

where $a$ is a constant.
• Dec 22nd 2012, 08:41 AM
jakncoke
Re: Nonlinear second order differential equation
Ok, This method works for all equations of the form $y'' = f(y', y)$.
Now Assume a new variable, v = y'. So $\frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}v}{\mathrm{d}y} \frac{\mathrm{d}y}{\mathrm{d}x}$

Since $\frac{\mathrm{d}y}{\mathrm{d}x}} = v$ This now becomes $= \frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}v}{\mathrm{d}y} v$
Now you have a Differential equation of the first order, $v \frac{\mathrm{d}v}{\mathrm{d}y} = g(v, y)$. For your specific equations turns into a seperable ODE,
$v \frac{\mathrm{d}v}{\mathrm{d}y} = \frac{a}{y} * v$ which turns into

$\mathrm{d}v} = \frac{a}{y} \mathrm{d}y$

So v = $v = ln(y) + k_0$

Now again Since $v = \frac{\mathrm{d}y}{\mathrm{d}x}$ which means $\frac{1}{ln(y)+k_0} \mathrm{d}y = \mathrm{d}x$
and you get some super ugly integral.
• Dec 22nd 2012, 11:00 AM
JulieK
Re: Nonlinear second order differential equation
Thank you
• Dec 23rd 2012, 12:06 AM
JJacquelin
Re: Nonlinear second order differential equation
Hi !

The obvious solutions are y(x)=constant.
The other solutions cannot be expressed as a combination of a finit number of elementary functions.
They involves some special functions such as li(x) or Ei(x) :
Logarithmic Integral -- from Wolfram MathWorld
Exponential Integral -- from Wolfram MathWorld
About special functions, pp.18-36 of the paper "Safari on the contry of special Functions" :
JJacquelin's Documents | Scribd
• Dec 26th 2012, 12:47 AM
tomjackson
Re: Nonlinear second order differential equation
Can someone please confirm that the above solution is OK? I tried a lot but I could not find the result. Is there any other way to solve this problem? I could not find any divisor for R[x].

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