Solving a second-order differential equation

[docnet]

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.

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.

 

[docnet]

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...

 

[docnet]

By using DSolve with initial value conditions,

$$\frac{d^2u}{dt^2}=\frac{1}{u^3}-\Big(\frac{u^4+1}{u^5}\Big)\Big(\frac{du}{dt}\Big)^2\Rightarrow u(t)=\sqrt{1+t^2}$$

 

[docnet]

(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