How to Estimate with a Computer program a 3rd order diff eq

How to Estimate with a Computer program a 3rd order differential equation with 4 variables and 4 equations? The equations are derived from the geodesics equation and the Schwarzchild metric. The variables are r, theta, phi and t. Any hints would be appreciated.

 

Equation 1:
[tex]\frac{d}{d\tau}(\frac{2r}{r-2m}(\frac{dr}{d\tau})) + \frac{2m}{(r-2m)^{2}}(\frac{dr}{d\tau})^{2} - r(\frac{d\theta}{d\tau})^{2} - rsin^{2}\theta (\frac{d\phi}{d\tau})^{2} + \frac{mc^{2}}{r^{2}}(\frac{dt}{d\tau})^{2} = 0[/tex]

Equation 2:
[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) - r^{2}sin\theta cos\theta (\frac{d\phi}{d\tau}) = 0[/tex]

Equation 3:
[tex]\frac{d}{d\tau}(r^{2}sin^{2}\theta (\frac{d\phi}{d\tau})) = 0[/tex]

Equation 4:
[tex]\frac{d}{d\tau}(\frac{r-2m}{r}(\frac{dt}{d\tau})) = 0[/tex]

where m and c are constants

 

Yes the equation is 2nd order not 3rd.

 

Just checking my math, in the first part of Equation 2:

[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = 2r\frac{dr}{d\tau}\frac{d\theta}{d\tau} + r^{2}(\frac{d\theta}{d\tau})^{2}[/tex]

Is this correct?

 

Any suggestions on how to solve for r and [tex]\theta[/tex] from the first two equations if the solution from equation 3 is :

[tex]\frac{d\phi}{d\tau} = \frac{constant3}{r^{2}sin^{2}\theta}[/tex]

and equation 4 is:

[tex]\frac{dt}{d\tau} = \frac{(constant4)r}{r-2m}[/tex]

 

tau takes the place of s in geodesic equations and is called proper time in a "local free-fall frame replacing s by:

[tex]ds^{2} = -c^{2}d\tau^{2}[/tex]