Find the differential equation or system of differential equations

[Askhwhelp]

Find the differential equation or system of differential equations ***

Find the differential equation or system of differential equations assoicated with the following flows
a) $\phi_t (x) = \frac{x}{\sqrt{1-2x^2t}}$ on ${\mathbb R}$

b) $\phi_t (x,y) = (xe^t, \frac{y}{1-y^t})$ on ${\mathbb R}^2$

The ways I solve these two questions are that I simply take the derivatives of them

for (a), $\left.\dfrac{d}{dt}\right|_{t=0} \phi_t (x)$ if this is the right way, check you check my answer, $\frac{-x}{2} - 2x^2$

for (b), $\left.\dfrac{d}{dt}\right|_{t=0} \phi_t (x,y)$ if this is the right way, check you check my answer, $(xe^t, \frac{ty}{(1-y^t)^2})$

 

[haruspex]

In (a), I assume that what you are looking for is a differential equation involving only ϕ, dϕ/dx and x.
If so, you need to differentiate wrt x, and you can't get rid of the t just by evaluating at t=0.

 

[Askhwhelp]

In (a), I assume that what you are looking for is a differential

Just edit my question...please take a look again to see if anything changes to your response

 

[haruspex]

Just edit my question...please take a look again to see if anything changes to your response

Yes, it means I'm a bit out of my depth.. but I think you need to bear in mind that x = x(t), and it's dϕ/dt, not ∂ϕ/∂t.

 

[Dick]

Yes, it means I'm a bit out of my depth.. but I think you need to bear in mind that x = x(t), and it's dϕ/dt, not ∂ϕ/∂t.

I'm not an expert in the subject either, but in flow problems like this I think you have $x(t)=\phi(x,t)$. Treat the x in $\phi(x,t)$ as a constant. And I'd be interested in how Askhwhelp got either solution. I think the first one is just plain wrong. And for another thing, they don't look like differential equations to me and the second one even has a t in it. How can that be if you set t=0?

 

[haruspex]

Having read up on this, I think I understand it well enough now to give a more helpful answer.

for (a), $\left.\dfrac{d}{dt}\right|_{t=0} \phi_t (x)$ my answer, $\frac{-x}{2} - 2x^2$

As Dick says, it should be partial differentiation, the result should be equated to dx/dt, and that answer is wrong.
Please post your working.