Hello, I'm trying to figure out how solve the below for varying time and/or corresponding theta. I'm stumped.. any insight is greatly appreciated. Thank you


$\ddot{R} = −\alpha|v|\dot{R}−\beta\_{s}\frac{|v|}{R}\dot{R}-\beta\_{c}\frac{g}{|v|}\dot{R} + [R\dot{\theta}^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta$


$\ddot{\theta} = −\alpha|v|\dot{\theta}−\beta\_{s}\frac{|v|}{R}\dot{\theta}-\beta\_{c}\frac{g}{|v|}\dot{\theta}- \frac{2\dot{R}\dot{\theta}}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta$


with $|v|=\sqrt{\dot{R}^2/cos^2\delta + R^2\dot{\theta}^2} $


where   

$\alpha = 0.00900350489819847$  

$\beta\_{s} = 0.0022919958783044724$  

$\beta{c}=0.0062192039382357378$  

$\delta=0.2574$  

$R=0.242$  

$g=9.807$


with the 'initial conditions' at t = 0:


$R(0) = R$  

$\dot{R}(0) = 0$  

$\theta(0) = 57.899552605659885$  

$\dot{\theta}(0) = 2.7371516764119637$


Per PaPiro's suggestion, I've converted the above to first order ODE's and got the following: 


Let $x1=\dot{R}$ and $\dot{x1}=\ddot{R}$ .


$x1=\dot{R}$   

$\dot{x1} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2} ) x1−\beta\_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2}}{R}x1-\beta\_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2}}x1 + $
  

              
$[R\dot{\theta}^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta = \ddot{R}$


Let $x2=\dot{\theta}$ and $\dot{x2}=\ddot{\theta}$ .


$x2=\dot{\theta}$   

$\dot{x2} = −\alpha(\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2} )x2−\beta\_{s}\frac{\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2}}{R}x2-\beta\_{c}\frac{g}{\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2}}x2-$
  

              
$ \frac{2\dot{R}x2}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta$


Per PaPiro's suggestion, I've rewritten the above $\dot{x1}$ as a function of x2 and the equation for $\dot{x2}$ as a function of x1: 


$x1=\dot{R}$   

$\dot{x1} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2x2^2} ) x1−\beta\_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2x2^2}}{R}x1-\beta\_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2x2^2}}x1 +$
  

              $[Rx2^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta = \ddot{R}$


$x2=\dot{\theta}$   

$\dot{x2} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2 x2^2} )x2−\beta\_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2 x2^2}}{R}x2-\beta\_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2 x2^2}}x2-$
  

              $ \frac{2 \cdot x1 \cdot x2}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta = \ddot{\theta}$


Please let me know if this is correct and if I'm on the right track. I'll try plugging this into 4th Order Runge-Kutta next.



How to solve this System of Second Order Differential Equations (DE) with Initial Value Problem (IVP)
Suggestions: write your second order differential equations as a system of first order differential equations and use the 4th order Runge-Kutta method to solve the resultant system of ODEs.


Google for *Runge-Kutta ODE systems*... 


**ADDED:** Also, You can use Simulink (Matlab) or Xcos (Scilab) and block diagrams to solve your differential equations. A very good reference is Close, Frederick, Newell, *Modelling and Analysis of Dynamic Systems*, Wiley, 3rd Ed., 2001. Chap. 4, **Block Diagram and Computer Simulations** is the key.


Google for *block diagram and computer simulation*...


