# Thread: Nonlinear second order differential equation

1. ## Nonlinear second order differential equation

What is the solution of the follwoing differential equation

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

where $\displaystyle a$ is a constant.

2. ## Re: Nonlinear second order differential equation

Ok, This method works for all equations of the form $\displaystyle y'' = f(y', y)$.
Now Assume a new variable, v = y'. So $\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}v}{\mathrm{d}y} \frac{\mathrm{d}y}{\mathrm{d}x}$

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

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

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

Now again Since $\displaystyle v = \frac{\mathrm{d}y}{\mathrm{d}x}$ which means $\displaystyle \frac{1}{ln(y)+k_0} \mathrm{d}y = \mathrm{d}x$
and you get some super ugly integral.

Thank you

4. ## 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

5. ## 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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