# How to solve this non separable ODE?

• I
• jk22
There is a circular reasoning going on here. The equation is wrong, so changing the variables doesn't help.f

#### 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 ?

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

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This does not help since F still depends on v.

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.

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

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

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

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

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##.

$$x'=Ax+Bct$$
$$ct'=Dx+Ect$$
$$c^2t'^2-x'^2=(E^2-B^2)c^2t^2-(A^2-D^2)x^2+2(DE-AB)xct$$
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
$$\phi = \psi$$
$$\tanh\phi=-\frac{v}{c}$$
$$E=A= \cosh \phi=\frac{1}{\sqrt{1-v^2/c^2}}$$
$$B=D= \sinh \phi=\frac{-v/c}{\sqrt{1-v^2/c^2}}$$

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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.

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 ?

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

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)##