- #1
Loren Booda
- 3,125
- 4
Please solve
r=Kt/((dr/dt)2-c2)
where r and t are variables, and K and c are constants.
r=Kt/((dr/dt)2-c2)
where r and t are variables, and K and c are constants.
Originally posted by himanshu121
no its not non linear
It is I order diff equation
Originally posted by Loren Booda
Bravo for your elegant solution, MathNerd. Excuse my ignorance, but is my original equation at the end of the day exactly solvable analytically between r and t?
I invite you to see "Booda's Theorem" on my website, http://www.quantumdream.net. The above problem derives from the mathematics of "Relativity's Complex Probability" on that page.
Agreed (whether I understand the derivation entirely or not).since c is an arbitrary constant then c2-->c without any loss of generality
What I meant by the function r not being single-valued in general for any t means that for any given value of t there are multiple values of r that satisfy the equation between r and t.Originally posted by Loren Booda
MathNerd,
Would you define "single-valued" as you used it in reference to nonlinearity?
Originally posted by Orion1
[tex]r = \frac{Kt} { \left( \frac{dr}{dt} \right)^2 - c^2}[/tex]
[tex]\left( \frac{dr}{dt} \right)^2 - c^2 = \frac {K t}{r}[/tex]
[tex]\left( \frac{dr}{dt} \right)^2 = \frac{Kt}{r} + c^2[/tex]
[tex]\frac{dr}{dt} = \sqrt{ c^2 + \frac{Kt}{r}}[/tex]
[tex]dr = \int \sqrt{ c^2 + \frac{Kt}{r}} dt[/tex]
differential solution:
[tex]r(t) = \frac{2r \left( c^2 + \frac{Kt}{r} \right)^{3/2}}{3K} + C[/tex]
Originally posted by Orion1
no other known solutions exist.
Originally posted by Orion1
[tex]dr = \int \sqrt{ c^2 + \frac{Kt}{r(t)}} dt[/tex]
differential solution:
[tex]r(t) = t \sqrt{ c^2 + \frac{Kt}{r(t)}} + C[/tex]
Originally posted by Orion1
[tex]r(t) \left( \frac{dr}{dt} \right)^2 = Kt + c^2 r(t)[/tex]
[tex]r(t) \left( \frac{dr}{dt} \right)^2 = Kt + c^2 \left( t \sqrt{ c^2 + \frac{Kt}{r(t)}} + C \right)[/tex]
Originally posted by Orion1
[tex]p = \sqrt{ \frac {K t} {r} + c }[/tex]
MathNerd Theorem:
[tex]
\int p ( \frac {a_1} {p+\sqrt{c}} + \frac {a_2} {p-\sqrt{c}} + \frac {a_3} {p-\epsilon_+} + \frac {a_4} {p-\epsilon_-} + \frac {a_5} {p-v} ) dp = \int \frac {dt} { 2 t }
[/tex]
Integral:
[tex]\int \frac{dt}{2t} = \frac{log(t)}{2} + C[/tex]
semi-differential solution:
[tex]\frac{log(t)}{2} + C = \int p ( \frac {a_1} {p+\sqrt{c}} + \frac {a_2} {p-\sqrt{c}} + \frac {a_3} {p-\epsilon_+} + \frac {a_4} {p-\epsilon_-} + \frac {a_5} {p-v} ) dp[/tex]
This equation is a differential equation that represents the relationship between the position (r) of an object, its velocity (dr/dt), and a constant (K) over time (t). The denominator of the equation, (dr/dt)2-c2, represents the speed (c) of the object, which is squared and subtracted from the square of the object's velocity. This equation is commonly used in physics and engineering to model the motion of objects.
To solve this differential equation, you can use techniques such as separation of variables, integration, or substitution. The specific method used will depend on the form of the equation and the initial conditions given. It is also possible to use computer software or numerical methods to approximate a solution.
This equation has many real-life applications in fields such as physics, engineering, and economics. It can be used to model the motion of objects, analyze electrical circuits, and predict population growth. It is also used in the study of fluid dynamics, thermodynamics, and other areas of science and engineering.
As with any mathematical model, there are limitations to this equation. It assumes that the object being studied has constant velocity and that the speed of the object remains constant over time. It also does not take into account external factors such as air resistance or friction. Additionally, the equation may not accurately model complex systems or situations with rapidly changing variables.
This equation can be modified by changing the constant (K) or the value of the speed (c). These modifications can be used to model different scenarios and situations. For example, in a scenario with air resistance, the speed (c) may be adjusted to account for the effect of air resistance on the object's motion.