Numerical approximations with series

readme.md

@@ -1,2 +1,10 @@
 # Dynamical systems examples Lynch
 Repository for examples related with book Dynamical system with applications using Python by Stephen Lynch. 
+
+# List of examples included
+## Chapter 2
+- program 2-a:
+- program 2-b:
+- program 2-c:
+- program 2-d:
+- program 2-e: Numerical and truncated series solutions. 

scripts/program-02d.py

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+# Program 02d: Power series solution of a second order ODE;
+# example 8, chapter 2
+from sympy import Function, dsolve, pprint
+from sympy.abc import t
+
+x = Function('x')
+dxdt2 = x(t).diff(t, 2)
+dxdt = x(t).diff(t)
+ode = dxdt2+2*t**2*dxdt+x(t)
+sol = dsolve(ode, hint='2nd_power_series_ordinary', n=6)
+print('Solution:\n')
+pprint(sol)

scripts/program-02e.py

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+# Program 02e: Numerical and truncated series solution
+# figure 2.6
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+
+
+def ode(X, t):
+    x = X[0]
+    y = X[1]
+    dxdt = y
+    dydt = x-t**2*y
+    return [dxdt, dydt]
+
+
+X0 = [1, 0]
+t = np.linspace(0, 10, 1000)
+sol = odeint(ode, X0, t)
+x = sol[:, 0]
+y = sol[:, 1]
+fig, ax = plt.subplots()
+ax.plot(t, x, label='Numerical')
+ax.plot(t, 1+t**2/2+t**4/24, 'r-', label='Truncated series')
+plt.xlabel('t')
+plt.ylabel('x')
+plt.xlim(0, 4)
+plt.ylim(0, 4)
+ax.legend()
+plt.show()