Trying to solve a transcendental differential equation

In summary, solving a transcendental differential equation involves finding a function that satisfies the equation and its derivatives, but does not have a closed-form solution. This requires numerical methods and approximations, such as Taylor series expansions, to find an approximate solution. Additionally, techniques such as separation of variables and substitution may be used to simplify the equation and make it more manageable for solving. However, solving these types of equations can be challenging and often requires advanced mathematical knowledge and computational skills.
  • #1
Portuga
56
6
Homework Statement
Solving a Brachistochrone curve with drag.
Relevant Equations
##\vec F = -k \vec v##, ##v = \frac{C}{\sqrt{1 + \left( y^\prime \right)^2}}##
Well, I followed the strategy used by A.S. Parnovsky in his article (\url{http://info.ifpan.edu.pl/firststep/aw-works/fsV/parnovsky/parnovsky.pdf}) and found this differential equation: $$-\frac{g x}{C^{2}} = -\frac{\beta^{2} {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta {y^{\prime}}^{2} \log\left({y^{\prime}}^{2} + 1\right) - 2 \, \beta {y^{\prime}}^{2} \log\left(\beta - {y^{\prime}}\right) - \beta^{3} - \beta^{2} {y^{\prime}} + \beta^{2} \arctan\left({y^{\prime}}\right) - {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta \log\left({y^{\prime}}^{2} + 1\right) - 2 \, \beta \log\left(\beta - {y^{\prime}}\right) - \beta - {y^{\prime}} - \arctan\left({y^{\prime}}\right)}{2 \, {\left(\beta^{2} + 1\right)}^{2} {\left({y^{\prime}}^{2} + 1\right)}},$$ where ##\beta, g## and ##C## are constants.
The problem is: how to proceed? This is clearly an transcendental equation for ##y^\prime## and the autor didn't solve it explicitly. He purposed the use of a series expansion of reciprocal powers, but I was imagining if this was the most reasonable way to do it.
 
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  • #2
Closed-form solutions are few and far between. Anything that gets a little complicated (and this example is very complicated) can not be expected to have a simple closed-form solution. I suspect that the use of a series expansion is the most practical way to get a closed-form approximation to any desired accuracy.
 
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  • #3
Portuga said:
but I was imagining if this was the most reasonable way to do it.

I believe the most reasonable way to handle this is to numerically solve the DE. However, before doing this, looks like he made a typo in eqn 35 in the denominator of integrand. Should it not be (where I use x for y'):
$$
\int \frac{x}{(1+x^2)^2(x-\beta)}dx
$$
And even when I just try to solve the integral in Mathematica, the answers do not agree:

Mathematica:
expA=Integrate[x /((1+x^2)^2(x-\[Beta])),x]//Apart
expB=(x+\[Beta])/(2(1+x^2)(1+\[Beta]^2))+
(1-\[Beta]^2)/(2(1+\[Beta]^2)^2) (ArcTan[x]-Pi/2)+
\[Beta]/(2(1+\[Beta]^2)^2) Log[(x-\[Beta])^2/(1+x^2)]

expA===expB

expA/.{x->1,\[Beta]->0.5}
expB/.{x->1,\[Beta]->0.5}

False

0.155785

-0.221206

Would have to resolve these issues first .
 
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Related to Trying to solve a transcendental differential equation

1. What is a transcendental differential equation?

A transcendental differential equation is an equation that involves both a function and its derivative, and the function itself cannot be expressed in terms of elementary functions like polynomials or trigonometric functions.

2. Why are transcendental differential equations difficult to solve?

Transcendental differential equations are difficult to solve because they do not have a general method for finding solutions. Each equation must be solved using specific techniques and methods, which can be complex and time-consuming.

3. Can a computer program be used to solve transcendental differential equations?

Yes, a computer program can be used to solve transcendental differential equations. However, the program must be specifically designed to handle these types of equations and may require advanced mathematical algorithms.

4. Are there any real-life applications of solving transcendental differential equations?

Yes, transcendental differential equations have many real-life applications in fields such as physics, engineering, and economics. They are often used to model complex systems and make predictions about their behavior.

5. What are some common techniques for solving transcendental differential equations?

Some common techniques for solving transcendental differential equations include separation of variables, substitution, and using power series or numerical methods. Each technique has its own advantages and is used depending on the specific equation being solved.

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