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Ch. 13.5 Solve ODE using scipy.integrate.odeint()

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Program 1

Solve ODE using scipy.integrate.odeint()

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 Mathematical Problem Formulation

                                        
        

    

        The differential equation is:
        $$ \frac{dx}{dt} = -k \cdot x $$
        $$ x(t = 0) = 10 $$
    




                                    

                                    
                                                                    

                            

                                                    

                    

                

                                

                    
                    

                        

                            Program 2
                            
Solve ODE using scipy.integrate.odeint()

                        

                        

                            
                            
                            
                        

                    


                    
                    

                        
                        

                            

                                 Python 3.11
                                Ready
                            

                            

                                
                                
                            

                            
                            

                                

                                    
                                    
                                    
                                    
                                

                                

                                    
                                

                            

                        


                        
                        

                            

                                
                                
                                                                
                                                            


                            
                            

                                

                                

                            


                            
                            

                                

                                    
Click "Run Code" to execute script and render figures.

                                

                            


                            
                                                        

                                

                                                                        

                                        
 Mathematical Problem Formulation

                                        
        
    

    
The ordinary differential equation (ODE) given by:

    \[
    \frac{dx}{dt} = -k \cdot x,
    \]
    where \( x(t) \) is a function of time \( t \) and \( k \) is a positive constant. Consider three different values of \( k \): \( k = 0.5, 1, 2.5 \) over the time interval \( t = [0, 10] \) with the initial condition \( x(0) = 10.0 \).

                                    

                                    
                                                                    

                            

                                                    

                    

                

                                

                    
                    

                        

                            Program 3
                            
Solve ODE using scipy.integrate.odeint()

                        

                        

                            
                            
                            
                        

                    


                    
                    

                        
                        

                            

                                 Python 3.11
                                Ready
                            

                            

                                
                                
                            

                            
                            

                                

                                    
                                    
                                    
                                    
                                

                                

                                    
                                

                            

                        


                        
                        

                            

                                
                                
                                                                
                                                            


                            
                            

                                

                                

                            


                            
                            

                                

                                    
Click "Run Code" to execute script and render figures.

                                

                            


                            
                                                        

                                

                                                                        

                                        
 Mathematical Problem Formulation

                                        
        
    

        The differential equation is:
        $$ \frac{d^2x}{dt^2} + \lambda \frac{dx}{dt} + kx = 0 $$
        $$ x_{t=0} = 0 \quad \text{and} \quad \left. \frac{dx}{dt} \right|_{t=0} = 0.7$$
    




                                    

                                    
                                                                    

                            

                                                    

                    

                

                            
            
            
            
            
            

                

                    

                         Theoretical Principles & Computational Physics Guide
                    

                    

                        Solve ODE using scipy.integrate.odeint(): Differential Equations & Dynamical Simulation                    

                    

                         Dr. Alorika Chatterjee & Dr. Anirban Shaw
                         Science & Physics Learners
                         Python 3 / NumPy / SciPy
                    

                


                
                

                    

                         1. Core Principles & Theoretical Framework
                    

                    

                        
                    

                        
 Dynamical Evolution: Solves differential equations governing kinematics, planetary orbits, electrical circuits, and wave propagation.

                        
 Step-by-Step Stepping: Advances continuous state vectors forward in discrete time intervals $\Delta t$ using slope estimates.

                        
 Conservation Laws: Evaluates preservation of physical invariants (total energy, angular momentum) across long integration horizons.

                    
                    

                


                
                

                    

                         2. Mathematical Formulation & Algorithmic Pipeline
                    

                    

                        
                    
Vector initial value problem and numerical state advancement:

                    
$$ \frac{d\mathbf{y}}{dt} = \mathbf{f}(t, \mathbf{y}), \quad \mathbf{y}(t_0) = \mathbf{y}_0 \implies \mathbf{y}_{n+1} = \mathbf{y}_n + \Delta t \cdot \mathbf{\Phi}(t_n, \mathbf{y}_n, \Delta t) $$
                    

                


                
                

                    

                         3. Numerical Mechanics & Implementation Logic
                    

                    

                        
                    
    
                        
  • Coupled Reduction: Convert $n$-th order ODEs into systems of $n$ coupled first-order equations.
    • 
                        
  • Intermediate Slopes: Compute weighted intermediate derivatives ($k_1, k_2, k_3, k_4$ in RK4).
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  • State Update: Advance state variables simultaneously and verify energy/phase invariants.
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                         4. Relevant Computational Practice Exercises
                    

                    

                                                    

                                Exercise 1
                                

                                    Solve the differential equation for Solve ODE using scipy.integrate.odeint() over 1000 time steps and plot state coordinates vs time.                                

                                
                            

                                                    

                                Exercise 2
                                

                                    Compare numerical solution stability for small vs large step sizes $h = \Delta t$.                                

                                
                            

                                                    

                                Exercise 3
                                

                                    Plot phase space trajectory demonstrating physical stability or damping attractors.                                

                                
                            

                                            

                


                
                

                     Standard Literature & References:
                    
    
                                                    
  • John R. Taylor, "Classical Mechanics", University Science Books (2005).
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  • Mark Newman, "Computational Physics", University of Michigan (2013).
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