- #1
Farina
- 39
- 0
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!
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!