cf. David Darmofal. 16.901 Computational Methods in Aerospace Engineering, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 22 May, 2016). License: Creative Commons BY-NC-SA

# Lecture 1 Numerical Integration of Ordinary Differential Equations: An Introduction


```python
import sympy
m_p, g = sympy.symbols('m_p g', real=True)
```


```python
t = sympy.Symbol('t',real=True)
```


```python
u = sympy.Function('u')(t)
D = sympy.Function('D')(u)
```


```python
SphFreeFall = sympy.Eq(m_p*sympy.diff(u,t),m_p*g - D)
sympy.pprint( SphFreeFall )
```

        d                         
    m_p⋅──(u(t)) = g⋅m_p - D(u(t))
        dt                        


For low speeds, 


```python
rho_g, mu_g = sympy.symbols("rho_g mu_g",real=True) 
# density and dynamic viscosity of the atmosphere, 
```


```python
a = sympy.symbols("a",real=True)
```


```python
Re = 2*rho_g*u*a/mu_g
```


```python
C_D = 24./Re + 6./(1. + sympy.sqrt(Re)) + 0.4
```


```python
Dlow = 0.5*rho_g*sympy.pi*a**2*u**2*C_D 
```


```python
Dlow

```




    0.5*pi*a**2*rho_g*(0.4 + 6.0/(sqrt(2)*sqrt(a*rho_g*u(t)/mu_g) + 1.0) + 12.0*mu_g/(a*rho_g*u(t)))*u(t)**2




```python
sympy.pprint( SphFreeFall.subs( D, Dlow) )
```

        d                   2     ⎛                 6.0               12.0⋅μ_g ⎞  
    m_p⋅──(u(t)) = - 0.5⋅π⋅a ⋅ρ_g⋅⎜0.4 + ───────────────────────── + ──────────⎟⋅u
        dt                        ⎜             ____________         a⋅ρ_g⋅u(t)⎟  
                                  ⎜            ╱ a⋅ρ_g⋅u(t)                    ⎟  
                                  ⎜      √2⋅  ╱  ──────────  + 1.0             ⎟  
                                  ⎝         ╲╱      μ_g                        ⎠  
    
    2           
     (t) + g⋅m_p
                
                
                
                



```python

```

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16-901-spring2005-mitocw/lectnotes.ipynb
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2021-12-24T16:00:41.000Z
16-901-spring2005-mitocw/lectnotes.ipynb
ernestyalumni/CompPhys
1f5d7559146a14a21182653b77fd35e6d6829855
Apache-2.0
2018-01-16T22:34:47.000Z
2019-01-29T22:37:10.000Z
16-901-spring2005-mitocw/lectnotes.ipynb
ernestyalumni/CompPhys
1f5d7559146a14a21182653b77fd35e6d6829855
Apache-2.0
2017-01-24T19:18:42.000Z
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Qwen/Qwen-72B
1. YES
2. YES