Solving multiple linear ordinary differential equations in SymPy

I am using Python 3.5 in Jupyter (formerly iPython). The original notebook is available at my github examples repository.

Presume we wish to solve the coupled linear ordinary differential equations given by

where \(u(t)\) is the step function and \(x(0)=5\) and \(y(0) = 10\).

%matplotlib inline

# import symbolic capability to Python- namespace is a better idea in a more general code.
from sympy import *

# print things all pretty
from sympy.abc import *
init_printing()

Need to define variables as symbolic for sympy to use them.

t, C1, C2= symbols("t C1 C2")
x, y =  symbols("x y", cls = Function, Function = True)

First we must define the governing equations as equalities or expressions. We have the option, I will do one each way for the sake of illustration, as opposed to simplicity.

eq1 = Eq(3 * diff(x(t), t), y(t))
eq1

\begin{equation*} 3 \frac{d}{d t} x{\left (t \right )} = y{\left (t \right )} \end{equation*}

I'd like to use the Heaviside function. It doesn't work, but I'm showing how I tried anyway.

import sympy.functions.special.delta_functions as sfuncs

eq2 = Eq(diff(y(t),t), - 3 * y(t) - 15 * x(t) + 4 * sfuncs.Heaviside(t))
# Note: Heaviside doesn't work for me in the solver, so I've left it here as "proper" but dokn't use it.
eq2 = Eq(diff(y(t),t), - 3 * y(t) - 15 * x(t) + 4 * 1)
eq2

\begin{equation*} \frac{d}{d t} y{\left (t \right )} = - 15 x{\left (t \right )} - 3 y{\left (t \right )} + 4 \end{equation*}

Solving the differential equations. The ics =... should apply the initial conditions. Doesn't work. Perhaps some day.

soln = dsolve((eq1, eq2), ics = {x: 5, y: 0})
soln

\begin{equation*} \left [ x{\left (t \right )} = \frac{1}{3} \left(C_{1} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + C_{2} \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) e^{- \frac{3 t}{2}} + \frac{4}{15}, \quad y{\left (t \right )} = \left(\left(- \frac{3 C_{1}}{2} - \frac{\sqrt{11} C_{2}}{2}\right) \sin{\left (\frac{\sqrt{11} t}{2} \right )} + \left(\frac{\sqrt{11} C_{1}}{2} - \frac{3 C_{2}}{2}\right) \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) e^{- \frac{3 t}{2}}\right ] \end{equation*}

Solving for the constants. I'm substituting t = 0, then the initial values for x and y. You'll note that the first equation doesn't need y(0) substituted, while the second doesn't need x(0) substituted. I found this out after and did that for brevity.

constants = solve((soln[0].subs(t,0).subs(x(0),5), soln[1].subs(t,0).subs(y(0),10)),{C1,C2})
constants

\begin{equation*} \left \{ C_{1} : \frac{313 \sqrt{11}}{55}, \quad C_{2} : \frac{71}{5}\right \} \end{equation*}

Let's put in our constants and see what we get. I'm using .rhs to pull out the right side of the solution. You can look at soln[0] to see what I mean, or try help(soln[0]) and read the results.

xsoln = expand(soln[0].rhs.subs(constants))
xsoln

\begin{equation*} \frac{4}{15} + \frac{313 \sqrt{11}}{165} e^{- \frac{3 t}{2}} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + \frac{71}{15} e^{- \frac{3 t}{2}} \cos{\left (\frac{\sqrt{11} t}{2} \right )} \end{equation*}

ysoln = soln[1].rhs.subs(constants)
ysoln

\begin{equation*} \left(- \frac{172 \sqrt{11}}{11} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + 10 \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) e^{- \frac{3 t}{2}} \end{equation*}

eq1.subs(x(t),xsoln).subs(y(t),ysoln)

\begin{equation*} 3 \frac{d}{d t}\left(\frac{4}{15} + \frac{313 \sqrt{11}}{165} e^{- \frac{3 t}{2}} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + \frac{71}{15} e^{- \frac{3 t}{2}} \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) = \left(- \frac{172 \sqrt{11}}{11} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + 10 \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) e^{- \frac{3 t}{2}} \end{equation*}

Is equation 1 true with this solution set?

Eq(simplify(3*diff(xsoln,t)),simplify(ysoln))

\begin{equation*} \mathrm{True} \end{equation*}

Is equation 2 true with this solution set?

Eq(simplify(diff(ysoln,t)),simplify(4-3*ysoln-15*xsoln))

\begin{equation*} \mathrm{True} \end{equation*}

So that's it. Out answers are indeed given by

Eq(x(t),xsoln)

\begin{equation*} x{\left (t \right )} = \frac{4}{15} + \frac{313 \sqrt{11}}{165} e^{- \frac{3 t}{2}} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + \frac{71}{15} e^{- \frac{3 t}{2}} \cos{\left (\frac{\sqrt{11} t}{2} \right )} \end{equation*}

and

Eq(y(t),ysoln)

\begin{equation*} y{\left (t \right )} = \left(- \frac{172 \sqrt{11}}{11} \sin{\left (\frac{\sqrt{11} t}{2} \right )} + 10 \cos{\left (\frac{\sqrt{11} t}{2} \right )}\right) e^{- \frac{3 t}{2}} \end{equation*}

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