(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!

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

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