Convert Exponential form of Complex Equation into polar form with Sympy, python

Question

I am trying to solve a differential equation with sympy module in python. This is what I did

from sympy import *


t = sympy.Symbol('t')      
k = sympy.Symbol('k')      
m = sympy.Symbol('m')      
x = sympy.Function('x')(t)     

GDE = Eq(m*x.diff(t, 2) + k*x, 0)

solution = dsolve(GDE, x)
display(solution)

Now, I am trying to get a solution in polar form with both sin, cos and complex term (i) like this picture. In this picture, w_n is sqrt(k/m)

I tried

solution.rewrite(sin) 

but it returns an expression with hyperbolic functions .rewrite(cos) also returns an expression with hyperbolic functions.

I also tried

solution.rewrite(sin, cos)

but it gives the initial answer.

Is there any way to convert this exponential form of complex form to polar form without hyperbolic functions?

Thank you

Answer 1

You should declare your symbols as real or positive etc. Otherwise by default SymPy assumes something like a complex number:

In [7]: import sympy

In [8]: from sympy import *
   ...: 
   ...: 
   ...: t = sympy.Symbol('t', real=True)
   ...: k = sympy.Symbol('k', positive=True)
   ...: m = sympy.Symbol('m', positive=True)
   ...: x = sympy.Function('x', real=True)(t)
   ...: 
   ...: GDE = Eq(m*x.diff(t, 2) + k*x, 0)

Then various simplifications will happen automatically:

In [9]: sqrt(-k/m)
Out[9]: 
ⅈ⋅√k
────
 √m 

In [10]: exp(t * sqrt(-k/m))
Out[10]: 
 ⅈ⋅√k⋅t
 ──────
   √m  
ℯ      

In [11]: exp(t * sqrt(-k/m)).rewrite(cos)
Out[11]: 
     ⎛√k⋅t⎞      ⎛√k⋅t⎞
ⅈ⋅sin⎜────⎟ + cos⎜────⎟
     ⎝ √m ⎠      ⎝ √m ⎠

In fact with the assumptions set dsolve will give the sin, cos form automatically:

In [12]: dsolve(GDE)
Out[12]: 
             ⎛√k⋅t⎞         ⎛√k⋅t⎞
x(t) = C₁⋅sin⎜────⎟ + C₂⋅cos⎜────⎟
             ⎝ √m ⎠         ⎝ √m ⎠

Without these assumptions set it is still possible to manipulate the expression into the sin/cos form but some force=True is needed to override the assumptions checking:

In [25]: s = solution.rhs; s
Out[25]: 
           _____             _____
          ╱ -k              ╱ -k  
    -t⋅  ╱  ───        t⋅  ╱  ─── 
       ╲╱    m           ╲╱    m  
C₁⋅ℯ             + C₂⋅ℯ           

In [26]: s = expand(s, force=True); s
Out[26]: 
                ___                  ___
               ╱ 1                  ╱ 1 
    -ⅈ⋅√k⋅t⋅  ╱  ─        ⅈ⋅√k⋅t⋅  ╱  ─ 
            ╲╱   m               ╲╱   m 
C₁⋅ℯ                + C₂⋅ℯ              

In [27]: s = s.rewrite(sin); s
Out[27]: 
   ⎛       ⎛         ___⎞      ⎛         ___⎞⎞      ⎛     ⎛         ___⎞      ⎛         ___⎞⎞
   ⎜       ⎜        ╱ 1 ⎟      ⎜        ╱ 1 ⎟⎟      ⎜     ⎜        ╱ 1 ⎟      ⎜        ╱ 1 ⎟⎟
C₁⋅⎜- ⅈ⋅sin⎜√k⋅t⋅  ╱  ─ ⎟ + cos⎜√k⋅t⋅  ╱  ─ ⎟⎟ + C₂⋅⎜ⅈ⋅sin⎜√k⋅t⋅  ╱  ─ ⎟ + cos⎜√k⋅t⋅  ╱  ─ ⎟⎟
   ⎝       ⎝     ╲╱   m ⎠      ⎝     ╲╱   m ⎠⎠      ⎝     ⎝     ╲╱   m ⎠      ⎝     ╲╱   m ⎠⎠

In [28]: s = expand(s).collect(s.atoms(sin, cos)); s
Out[28]: 
             ⎛         ___⎞                     ⎛         ___⎞
             ⎜        ╱ 1 ⎟                     ⎜        ╱ 1 ⎟
(C₁ + C₂)⋅cos⎜√k⋅t⋅  ╱  ─ ⎟ + (-ⅈ⋅C₁ + ⅈ⋅C₂)⋅sin⎜√k⋅t⋅  ╱  ─ ⎟
             ⎝     ╲╱   m ⎠                     ⎝     ╲╱   m ⎠

In [29]: s = powsimp(s, force=True); s
Out[29]: 
             ⎛         ___⎞                     ⎛         ___⎞
             ⎜        ╱ 1 ⎟                     ⎜        ╱ 1 ⎟
(C₁ + C₂)⋅cos⎜√k⋅t⋅  ╱  ─ ⎟ + (-ⅈ⋅C₁ + ⅈ⋅C₂)⋅sin⎜√k⋅t⋅  ╱  ─ ⎟
             ⎝     ╲╱   m ⎠                     ⎝     ╲╱   m ⎠

In [30]: s = powsimp(s, force=True, deep=True); s
Out[30]: 
             ⎛      ___⎞                     ⎛      ___⎞
             ⎜     ╱ k ⎟                     ⎜     ╱ k ⎟
(C₁ + C₂)⋅cos⎜t⋅  ╱  ─ ⎟ + (-ⅈ⋅C₁ + ⅈ⋅C₂)⋅sin⎜t⋅  ╱  ─ ⎟
             ⎝  ╲╱   m ⎠                     ⎝  ╲╱   m ⎠