(adsbygoogle = window.adsbygoogle || []).push({}); Apparently this 2nd-order ODE has 3 solutions??

The following apparently has 3 solultions:

[tex]

\frac {d^2u}{d\theta^2} + u = -\frac {1}{ml^2u^2}f(u^{-1})

[/tex]

where:

u = 1/r

m = mass

l = angular momentum

One of the solutions is:

[tex]r=r_0e^{k\theta} \text{ where } \theta \text { varies logarithmically with time}[/tex]

Apparently there are also 2 additional solutions (depending on the value of the constant [tex]\alpha[/tex])

that could be in the form of:

[tex]r=Ae^{\sqrt{\alpha x}}+Be^{-\sqrt{\alpha x}} \text{ or }[/tex]

[tex]r=A\theta + B \text{ or }[/tex]

[tex]r=Asin({\sqrt{\alpha x}})+Bcos({\sqrt{\alpha x}})[/tex]

So, knowing:

[tex]

\frac {d^2u}{d\theta^2} + u = -\frac {1}{ml^2u^2}f(u^{-1})

[/tex]

and

[tex]r=r_0e^{k\theta}[/tex]

How does one specifically determine the equations of the additional solutions?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Apparently this 2nd-order ODE has 3 solutions?

**Physics Forums | Science Articles, Homework Help, Discussion**