Solving a second-order differential equation

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  • #26
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Dear all,

There is another error in our differential equation.

Starting over from part e) we solve the following equation for (dv/dt)2.
Screen Shot 2021-01-05 at 8.56.35 PM.png


giving
Screen Shot 2021-01-05 at 9.00.01 PM.png


and substitute it in the following equation
Screen Shot 2021-01-05 at 8.55.34 PM.png


giving
Screen Shot 2021-01-05 at 9.02.59 PM.png


Which offers no clear way to solve. We examine the following expression from part d).
Screen Shot 2021-01-05 at 8.58.25 PM.png


which makes
Screen Shot 2021-01-05 at 9.32.04 PM.png



which implies
Screen Shot 2021-01-05 at 9.33.10 PM.png



for some constant c. solving,
Screen Shot 2021-01-05 at 9.33.46 PM.png



substitute into the first equation to get
Screen Shot 2021-01-05 at 9.37.44 PM.png


which makes

Screen Shot 2021-01-05 at 9.44.39 PM.png


I think this way is not easy.
 
Last edited:
  • #27
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The expression

Screen Shot 2021-01-05 at 10.03.43 PM.png


has the solution

Screen Shot 2021-01-05 at 10.07.42 PM.png


and

Screen Shot 2021-01-05 at 10.11.35 PM.png
 
  • #28
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For the v component of the geodesic, we plug in
Screen Shot 2021-01-05 at 10.07.42 PM.png


into the unit speed equation
Screen Shot 2021-01-05 at 10.22.15 PM.png


giving
Screen Shot 2021-01-05 at 10.19.27 PM.png


and
Screen Shot 2021-01-05 at 10.20.36 PM.png


We then integrate to give
Screen Shot 2021-01-05 at 10.20.45 PM.png


and
Screen Shot 2021-01-05 at 10.25.52 PM.png


which is incorrect judging from the following plot in the uv plane

Screen Shot 2021-01-05 at 10.37.18 PM.png


but I think we are getting closer to the correct solution.
 
Last edited:
  • #30
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What course is this? Differential geometry?
 
  • #31
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You seem to have squared part of ##\frac{du}{dt}## twice. The left hand term above should reduce to 1/2.

I am sorry I should apologize because the solution to our ODE is not a geodesic, because we are solving the wrong ODE.

Not solving this problem has been eating me up. So, I revisited our previous calculations tonight, made sure to be extra careful not to make errors and ended up with the following. I still have not learned how to write LaTex equations, so these were typed in Microsoft Word, and these will not show up in quoted messages.

Screen Shot 2021-01-12 at 1.05.03 AM.png

Screen Shot 2021-01-12 at 1.05.11 AM.png
Screen Shot 2021-01-12 at 1.05.15 AM.png


I am unable to solve this nonlinear second order ODE by any method I am familiar with.

Is anyone able to compute it using Mathematica or Wolfram Αlpha?

Thank you.
 
Last edited:
  • #32
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Acccording to DSolve in Mathematica, the fearful ODE $$\frac{d^2u}{dt^2}=\frac{1}{u^3}-\Big(\frac{u^4+1}{u^5}\Big)\Big(\frac{du}{dt}\Big)^2$$
has "closed form" solutions given by...

Screen Shot 2021-02-23 at 4.49.41 PM.png
 
  • #33
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By using DSolve with initial value conditions,

$$\frac{d^2u}{dt^2}=\frac{1}{u^3}-\Big(\frac{u^4+1}{u^5}\Big)\Big(\frac{du}{dt}\Big)^2\Rightarrow u(t)=\sqrt{1+t^2}$$
 
  • #34
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(g) we plug ##u(t)## into the unit speed equation to find
$$\frac{dv}{dt}=\Big(\frac{t^4+t^2+1}{(1+t^2)^3}\Big)^{\frac{1}{2}}$$

which is a separable equation, when trying to compute
$$v(t)=\int \Big(\frac{t^4+t^2+1}{(1+t^2)^3}\Big)^{\frac{1}{2}}dt$$

we cannot find an explicit closed form solution
 

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