How to rearrange for a variable with SymPy

Question

I have a complex equation that I would like to rearrange for a variable vz(t) = ....

What is the most elegant way in SymPy ? I have determined the upper equation as a solution of an ODE:

import sympy as sym
sym.init_printing()
t,m,k,g,v0,psi = sym.symbols("t,m,k,g,v_0,psi")

vz = sym.Function('v_z')


eqn = sym.Eq(-k*vz(t)**2 - m*vz(t).diff(t) - m*g,0)

sol = sym.dsolve(eqn, ics = {vz(0): v0*sym.sin(psi)})

score 2 · Accepted Answer · answered Aug 18 '21 at 15:12

The solution you show can be simplified into solving involving atan and then rearranged. Some assumptions are needed on the symbols to make that rearrangement possible though. Actually if you set all symbols to be positive then you'll get a simpler result from dsolve in the first place:

In [51]: import sympy as sym
    ...: sym.init_printing()
    ...: t,m,k,g,v0,psi = sym.symbols("t,m,k,g,v_0,psi", positive=True)
    ...: 
    ...: vz = sym.Function('v_z', positive=True)
    ...: 
    ...: 
    ...: eqn = sym.Eq(-k*vz(t)**2 - m*vz(t).diff(t) - m*g,0)
    ...: 
    ...: sol = sym.dsolve(eqn, ics = {vz(0): v0*sym.sin(psi)})


In [52]: 

In [52]: sol
Out[52]: 
                  ⎛      ⎛           ⎛√k⋅v₀⋅sin(ψ)⎞⎞⎞
                  ⎜      ⎜       atan⎜────────────⎟⎟⎟
                  ⎜      ⎜  t        ⎝   √g⋅√m    ⎠⎟⎟
         √g⋅√m⋅tan⎜√g⋅√k⋅⎜- ── + ──────────────────⎟⎟
                  ⎝      ⎝  √m         √g⋅√k       ⎠⎠
v_z(t) = ────────────────────────────────────────────
                              √k

Introducing a parameter α = sqrt(m*g/k) this becomes:

In [65]: sol.subs(sqrt(k), sqrt(m)*sqrt(g)/alpha).expand()
Out[65]: 
              ⎛    ⎛v₀⋅sin(ψ)⎞   g⋅t⎞
v_z(t) = α⋅tan⎜atan⎜─────────⎟ - ───⎟
              ⎝    ⎝    α    ⎠    α ⎠

How to rearrange for a variable with SymPy

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