Inexact differential equation with unknown function M(t)

[tom.stoer]

I am looking for a general expression for an integrating factor μ(x,t) to solve the following diffential equation for x(t)

$$\frac{dx}{dt} = \frac{x - f}{x}$$

f = f(t) is an arbitrary function of t with f > 0 and df/dt < 0

Any ideas?

 

[JJacquelin]

This ODE is an Abel's equation which is not solvable on it's general form. The function f must be explicitely defined for any further attempt to solve it.

 

[MathematicalPhysicist]

JJacqulin, is there a proof that you cannot solve it for general good profile for $f(t)$?

 

[tom.stoer]

Thanks for the hint regarding the type of the equation. One can set z=1/x which transforms the equation in an Abel's equation of the 1st kind (in x it's 2nd kind). I found several statements that this type of equation is not solvable via integration in general, but that there are special cases where this may be possible - unfortunately w/o any further explanations what these special cases are :-(

 

[pasmith]

JJacqulin, is there a proof that you cannot solve it for general good profile for $f(t)$?

I think "cannot solve" here means "cannot solve analytically", not "can prove that a solution does not exist even for well-behaved $f$".

 

[JJacquelin]

"An Abel ordinary differential equation class generalizing known integrable classes" : E.S. Cheb-Terrah, A.D.Roche, arXIV:math/0002059v3, 23 Feb 2004
Note: in is impossible to say that a today non-integrable ODE wili not become integrable in the futur. All depends of the standard special functions defined in the present and in the futur. A new special function could be studied, published, implemented in the maths sofwares, accepted and recorded as a standard special function. Eventualy, this new special function may allow to solve an Abel's ODE which was not solvable before.
http://fr.scribd.com/doc/14623310/S...tions-Safari-au-pays-des-fonctions-speciales-

 

[JJacquelin]

Of course, "cannot solve" here means "cannot solve analytically". The ODE considered by tom.stoers is solvable thanks to numerial methods.

 

[tom.stoer]

I found several statements that this type of equation is not solvable via integration in general ...

What I mean is that the equation

$$\frac{dx}{dt} = \frac{x - f}{x}$$

cannot be solved via a general formula like the formula used for exact differential equations. This is what I am looking for and what is not known today, as I had to learn.

 

[Ben Niehoff]

If $f$ is a constant, you can solve this in terms of the Lambert W function. I discovered this by plugging it into Mathematica DSolve, but here's a simple way to get the answer. Consider

$$t = x - a + a \ln (x - a) + C$$
Then

$$\frac{dt}{dx} = 1 + \frac{a}{x - a} = \frac{x}{x - a}$$
which is what we want, so just invert $t(x)$:

$$\tilde{C} e^{t/a} = \frac{(x - a)}{a} e^{(x - a)/a}$$
or

$$x = a \big[ 1 + W( \tilde{C} e^{t/a} ) \big]$$
Edit: But I guess if $f$ is a constant, the equation was separable anyway.

 

[tom.stoer]

for f = const. the equation is separable, so no post is necessary ;-)