# Thread: Help with Differential Equation

1. ## Help with Differential Equation

Hello all,

I'm trying to solve a differential equation derived from Boyle's Gas Law which is setup as y' = f(x,x',y) and I can't figure out how to start.

The equation is below:

$\displaystyle dV/dt = -V/P * dP/dt$

I'm really lost on this one because of the differential of two different variables.

Any information you guys could provide would be really helpful.

Thanks,
Mitch

2. ## Re: Help with Differential Equation

You can write the ODE as

$\displaystyle \dfrac{1}{V} \dfrac{dV}{dt} + \dfrac{1}{P} \dfrac{dP}{dt} = 0$

$\displaystyle \dfrac{d \ln V}{dt} + \dfrac{d \ln P}{dt} = 0$

or

$\displaystyle \dfrac{d (\ln V + \ln P) }{dt} = 0$ or more conveniently $\displaystyle \dfrac{d \ln PV }{dt} = 0$.

This can now be integrated

3. ## Re: Help with Differential Equation

What I would do is write $\displaystyle \frac{dV}{dt}= -\frac{V}{P}\frac{dP}{dt}$ in 'differential form':
$\displaystyle dV= -\frac{V}{P}dP$ and then as

$\displaystyle \frac{dV}{V}= -\frac{dP}{P}$
and integrate both sides.

Of course, this gives exactly the same answer as Jester's suggestion.