1. ## Ordinary Differential equation

A uniform chain of total length 8 metres, settle 1 metre above the ledge and 2 metres hang on the other side. If x represents the length. The motion of the chain is

$(x+1)v(dv/dx) + v^2 = (x-1)g$

where v is velocity and g is a gravitational constant
show that by making the substitution $u = v^2$ we obtain
${du}/{dx} + (^{2}/_{x+1})u=2(^{x-1}/_{x+2})g$

now do i have to integrate this in order to subsitute?
any help on this would be appreciated as i have no idea where to start cheers people!.

2. Why not use an integrating factor?

$\phi (x) = e^{\int p(x)dx}$

$\phi (x) u(x) = \int \phi (x) r(x) dx$

Where

$\frac{du}{dx} + p(x)u = r(x)$

Is the format.

Sorry, the above is what you would do to solve after substitution. You don't need to integrate BEFORE you substitute.

I'm fairly sure your of the latter equation should be $2 (\frac{x-1}{x+1})g$, no?

If that's the case:

Divide through by $x+1$

After that apply the following considerations:

$u = v^2$

$\frac{du}{dv} = 2v$

$vdv = \frac{1}{2}du$

$\frac{dv}{dx}v = \frac{1}{2}\frac{du}{dx}$

Should help!

3. ## thanks

brilliant cheers got it!