How do I draw graph in sympy? - sympy

Question

I'm high school student, and I'm writing a report about profile velocity.
I don't know much about differential equations and Python, but I have to use both of them.
I'm trying to induce the velocity from (ma = mg - kv), and caculate a and s from v.
I caculated v successfully, but I have few questions.
import sympy
init_printing()
%matplotlib inline
(m, g, k, t) = symbols('m g k t')
v = Function('v')
deq = Eq( m*v(t).diff(t), m*g - k*v(t) )
eq = dsolve( deq, v(t) )
C1 = Symbol('C1')
C1_ic = solve( eq.rhs.subs( {t:0}), C1)[0]
r = expand(eq.subs({C1:C1_ic}))
the simple way to caculate C1 doesn't work
v(0) = 0
so I write
eq = dsolve( deq, ics={v(0):0})
but it has same result with
eq = dsolve( deq, v(t) )
how to caculate acc and draw a graph?
I try this code, but it doesn't work
a = diff(r, t)
r = dsolve( a, v(t))
r.subs({m:1, g:9.8, k:1})
plot( r , (t,0,100))

Answer 1

I don't get the same result from eq = dsolve( deq, ics={v(0):0}). Also you should declare m, g and k with positive=True.
In [50]: m, g, k = symbols('m g k', positive=True)
In [51]: t = Symbol('t')
In [52]: v = Function('v')
In [53]: deq = Eq( m*v(t).diff(t), m*g - k*v(t) )
In [54]: deq
Out[54]:
d
m⋅──(v(t)) = g⋅m - k⋅v(t)
dt
In [55]: dsolve(deq, v(t))
Out[55]:
k⋅(C₁ - t)
──────────
m
g⋅m + ℯ
v(t) = ─────────────────
k
In [56]: dsolve(deq, v(t), ics={v(0):0})
Out[56]:
⎛ m⋅log(g) m⋅log(m) ⅈ⋅π⋅m⎞
k⋅⎜-t + ──────── + ──────── + ─────⎟
⎝ k k k ⎠
────────────────────────────────────
m
g⋅m + ℯ
v(t) = ───────────────────────────────────────────
k
In [57]: sol = dsolve(deq, v(t), ics={v(0):0}).rhs
In [58]: sol.expand()
Out[58]:
-k⋅t
─────
m
g⋅m g⋅m⋅ℯ
─── - ──────────
k k
In [59]: factor_terms(sol.expand())
Out[59]:
⎛ -k⋅t ⎞
⎜ ─────⎟
⎜ m ⎟
g⋅m⋅⎝1 - ℯ ⎠
────────────────
k
You can compute and plot the acceleration like
In [62]: sol = factor_terms(sol.expand())
In [64]: a = sol.diff(t)
In [65]: a = sol.diff(t).subs({m:1, g:9.8, k:1})
In [66]: a
Out[66]:
-t
9.8⋅ℯ
In [67]: plot(a, (t, 0, 100))