General solution We consider the differential equation \[\frac{\dd y}{\dd t} = t\,y\] Using the method of separation of variables, you can find a general solution \[y(t)=c\cdot e^{\frac{1}{2}t^2}\] for some constant \(c\). Python can also determine this solution through the dsolve function in the sympy package:

>>> import sympy as sy    # load the sympy package and use an acronym
>>> t = sy.symbols('t')   # make a symbol t 
>>> y = sy.Function('y')  # make a symbolic function y
>>> ODE = sy.Eq(sy.Derivative(y(t),t), t*y(t))   # the ODE 
>>> ysol = sy.dsolve(ODE, y(t)); ysol   # determine the solution of the ODE 
y(t) = C1*exp(t**2/2))

Initial Value Problem In the dsolve command you also specify the initial value as an equation. We take as an example the initial value problem \[\frac{\dd y}{\dd t} = y^2,\quad y(0)=1\] Using the method of separation of variables, you can find a general solution \[y(t)=\frac{1}{c-t}\] for some constant \(c\). From \(y(0)=1\) follows that \(c=1\). Python can also determine this solution via the dsolve function:

>>> import sympy as sy   # load the sympy package and use an acronym 
>>> sy.init_printing()   # just for nicer display
>>> t = sy.symbols('t')  # make a symbol t 
>>> y = sy.Function('y') # make a symbolic function y
>>> ODE = sy.Eq(sy.Derivative(y(t),t), y(t)**2)   # the differential equation
>>> ysol = sy.dsolve(ODE, y(t), ics = {y(0):1})   # solve the initial value problem
>>> ysol
 
        -1 
y(t) = -----
       t - 1

>>> gensol = sy.dsolve(ODE, y(t))   # determine general solution
>>> constant = sy.solve([gensol.rhs.subs(t,0)-1])   # determine the constant 
>>> ysol = gensol.subs(constant); ysol 
 
         -1 
 y(t) = ----- 
        t - 1

ODE of order 2 The example is the initial value problem \[\frac{\dd^2y}{\dd t^2} -3\frac{\dd y}{\dd t}+ 2y = 0, \quad y(0)=-1,\quad \frac{dy}{dt}(0) = -3\] In the next section we will learn how to solve this problem, but the solution is \[y(t)=e^t - 2e^{2t}\] In the program below, we have use an alternative to specify derivatives in the differential equation and the initial conditions

>>> import sympy as sy   # load the sympy package and use an acronym  
>>> sy.init_printing()   # just for nicer display
>>> t = sy.symbols('t')  # make a symbol t
>>> y = sy.Function('y') # make a symbolic function y 
>>> ODE = sy.Eq(y(t).diff(t,2)-3*y(t).diff(t) + 2*y(t), 0); ODE 
 
                        2 
           d           d 
2*y(t) - 3*--(y(t)) + ---(y(t)) = 0 
           dt           2 
                      dt 

>>> ysol = sy.dsolve(ODE, y(t), ics = {y(0):-1, sy.diff(y(t),t).subs(t,0):-3}); ysol

            2*t   t
y(t) = - 2*e   + e 

>>> gensol = sy.dsolve(ODE, y(t)); genopl   # the general solution

       /         t\  t
y(t) = \C1 + C2*e /*e 

>>> # define equations for initial values and solve them. 
>>> constants = sy.solve([gensol.rhs.subs(t,0)+1, gensol.rhs.diff(t).subs(t,0)+3]) 
>>> ysol = gensol.subs(constants); ysol   # substitution of the values found

       /     t   \  t
y(t) = \- 2*e + 1/*e 

>>> sy.expand(ysol)   # expand brackets and simplify
 
             2*t   t 
 y(t) = - 2*e   + e

We continue this example to show how you can compute further with the symbolic solution in numerical sense. Substituting values of \(t\) does not work straightforwardly because ysol is a symbolic expression. Use lambdify to construct a function from the symbolic expression for \(y(t)\).

>>> yfun = sy.lambdify(t, yopl.rhs, "math")
>>> list(map(yfun, [0,1,2]))
[-1.0, -12.059830369402254, -101.80724396735783]