Can Non-Separable ODEs Be Solved with Coordinate Transformations?

[jk22]

I fell upon such an equation :

$$-E'(v)a(1+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$

It's not separable in E on one side and v expression on the other.

So I'm looking for methods to solve this maybe changes of coordinates ?

 

[mitochan]

Assuming E(v) is positive we can transform it to
$$-FdE=dv$$
where
$$F=\frac{a(1+\frac{c}{\sqrt{1-E^{-2}}})}{E(v+c\sqrt{1-E^{-2}})}$$

 

[jk22]

This does not help since F still depends on v.

 

[mitochan]

Ah, you are right.
Then I try
$$E=\coth w$$ and get
$$a(\sinh w +c\cosh w)\frac{dw}{dv}+v\cosh w+c\sinh w=0$$
It is not separable yet.

E should be dimensionless. Are a,c and v are also dimensionless parameters? If not I am afraid there is something wrong in your equation.

 

[jk22]

You're right this equation is wrong. I'll look at the calculation again.

 

[jk22]

It shall read :

$$-E'(v)*t*a(v+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$

But it anyhow is wrong since there shall be derivatives towards v and t.

Thanks for your help.

 

[mitochan]

If you explain physical meanings of E, v, a, t ,c and v, I would be able to consider more to help you.

 

[jk22]

E is supposed to be an analog to the gamma factor, t time, a acceleration, c speed of light and v speed.

But the derivation of the equation is wrong. I wrote a coordinate transformation $x'=A(v,t)x+B(v,t)t$, $t'=D(v,t)x+E(v,t)t$

The equations are $c^2t'^2-x'^2=c^2t^2-x^2$ and $\frac{dx'}{dt'}|_{x=0}=-v$

A mistake I made came while computing $dx'$ only with a partial derivative towards $v$.

 

[mitochan]

$$x'=Ax+Bct$$
$$ct'=Dx+Ect$$
$$c^2t'^2-x'^2=(E^2-B^2)c^2t^2-(A^2-D^2)x^2+2(DE-AB)xct$$
So
$$E=\cosh \phi$$
$$B=\sinh \phi$$
$$A=\cosh \psi$$
$$D= \sinh \psi$$

$$\frac{x'}{ct'}|_{x=0}=\frac{B}{E}=-\frac{v}{c}$$
$$\frac{x}{ct}|_{x'=0}=-\frac{D}{A}=\frac{v}{c}$$
So
$$\phi = \psi$$
$$\tanh\phi=-\frac{v}{c}$$
$$E=A= \cosh \phi=\frac{1}{\sqrt{1-v^2/c^2}}$$
$$B=D= \sinh \phi=\frac{-v/c}{\sqrt{1-v^2/c^2}}$$

 

[jk22]

Nice, I took a wrong start with writing dependencies of $A,B,D,E$ wrt v and t and then taking the differential $dx'$, making the equations unsolvable.

 

[zinq]

The question is explained so vaguely that it's impossible to tell whether a is a function or a constant.

 

[jk22]

But is it a problem that A is a function of v but v is a constant. So when we plot A(v), v varies, but it varies in function of what ? Human free choice or it could be safe to say that the unknowns depends on the known or given variables ?

 

[jim mcnamara]

Could you please tell us the exact source for this problem?

 

[jk22]

I think I got knotted in a circular reasoning :

If B(v), then allowing to change v (for any kind of ground) induces a dv/dt, then the hypothesis was wrong and in fact B(a,v), aso, such that the ODE is of infinite degree since $B(\{\frac{d^n v}{dt^n}\}_{n=0}^\infty)$