Thread: [SOLVED] Nonlinear second order ODE - help needed desperately!

Hey guys,

I was wondering whether anybody could help me with this problem:

I have the following nonlinear second order ODE:

$\displaystyle
(d^2y)/(dx^2) = -((dy/dx)^2)/y
$

and i need to find the true solution given the boundary conditions y(1) = 2.77 and y(2) = 4.23.

Now when i asked my lecturer for help he said the only advice he can give me is to re-write the differential equation so that everything on the left hand side equals zero.

so now i've got the following:

$\displaystyle
(d^2y)/(dx^2) + ((dy/dx)^2)/y = 0
$

And then he said to look back the product rule and work “backwards”.

But this have confused me even more so could anybody shed any light on this and possibly tell me how i am supposed to find the true solution?

Thank you for any help

(P.S. I know this is also posted in the Calculus section but i didn't realised this area was here so i apologise for that)

$\displaystyle
\frac{(d^2y)}{(dx^2)} + \frac{((\frac{dy}{dx})^2)}{y} = 0
$

can also be written:

$\displaystyle
y\frac{(d^2y)}{(dx^2)} + (\frac{dy}{dx})(\frac{dy}{dx}) = 0
$

In this form it is clear that this is the product rule in action:

$\displaystyle
\frac{d}{dx}(y \frac{dy}{dx}) = 0
$

So the derivative of the product of y with it's derivtive (with respect to x) is 0 and therefore the product of y with its derivative is a constant.

Wow thanks, that makes soooo much more sense than i was making of it! :-D lol

Can i just clarify something?

Does:

$\displaystyle
(\frac{dy}{dx})(\frac{dy}{dx}) = 0
$

or have i got myself confused again??? Coz i dont fully understand how you got to

$\displaystyle
\frac{d}{dx}(y \frac{dy}{dx}) = 0
$

sorry! :-P lol

Nevermind...i've managed to figure it out! :-P Thanks again! :-D