q_9_BVP_Updated code

Q_9_B_boundary_value_problem.py

@@ -2,28 +2,29 @@
 from scipy.integrate import solve_bvp
 import matplotlib.pyplot as plt
 
+# Define the function representing the differential equation
 def ode(x, y):
-    y1, y2 = y  #y1=y  ,y2=y'
-    dydx = [y2, y2 * np.cos(x) - y1 * np.log(y1)]  #define derivitive
-    return dydx
-# define boundary condition
+    return np.vstack((y[1], y[1]*np.cos(x) - y[0]*np.log(y[0] + 1e-10)))  # Add small positive offset
+
+# Define the boundary conditions
 def bc(ya, yb):
     return np.array([ya[0] - 1, yb[0] - np.exp(1)])
 
-# Define the range of x values where we want to solve the ODE
+# Define the x values for integration
 x = np.linspace(0, np.pi/2, 100)
 
-# Initial guess for the solution
+# Adjusted initial guess to ensure positivity of y1
 y_guess = np.zeros((2, x.size))
+y_guess[0] = 1  # Initialize y1 with a positive value
 
 # Solve the boundary value problem
 sol = solve_bvp(ode, bc, x, y_guess)
 
 # Plot the solution
-plt.plot(sol.x, sol.y[0], label='y(x)')
+plt.plot(sol.x, sol.y[0], label='Solution')
 plt.xlabel('x')
 plt.ylabel('y')
-plt.title('Solution of y\'\' = y\' * cos(x) - y * ln(y)')
+plt.title("Solution of y'' = y'cos(x) - yln(y)")
+plt.grid(True)
 plt.legend()
-plt.grid()
 plt.show()