(3x-1)y''-(3x+2)y'+(6x-8)y=0, y(0)=2, y'(0)=3

def g(y,x):
    y0 = y[0]
    y1 = y[1]
    y2 = (6*x-8)*y0/(3*x-1)+(3*x+2)*y1/(3*x-1)
    return [y1,y2]

init = [2.0, 3.0]
x=np.linspace(-2,2,100)
sol=spi.odeint(g,init,x)
plt.plot(x,sol[:,0])
plt.show()

import numpy as np
import scipy as sp
from scipy.integrate import odeint
import matplotlib.pyplot as plt

def g(y, x):
    y0 = y[0]
    y1 = y[1]
    y2 = ((3*x+2)*y1 + (6*x-8)*y0)/(3*x-1)
    return y1, y2

# Initial conditions on y, y' at x=0
init = 2.0, 3.0
# First integrate from 0 to 2
x = np.linspace(0,2,100)
sol=odeint(g, init, x)
# Then integrate from 0 to -2
plt.plot(x, sol[:,0], color='b')
x = np.linspace(0,-2,100)
sol=odeint(g, init, x)
plt.plot(x, sol[:,0], color='b')

# The analytical answer in red dots
exact_x = np.linspace(-2,2,10)
exact_y = 2*np.exp(2*exact_x)-exact_x*np.exp(-exact_x)
plt.plot(exact_x,exact_y, 'o', color='r', label='exact')
plt.legend()

plt.show()