The given nonlinear second-order ordinary differential equation (ODE) is:

$x^2 y'' + 2xy' - y + y^3 = (x - 3) + e^{y'} + (x + 3)^3 - e$

with initial conditions:

$y(1) = 4, \quad y'(1) = 1.$

Solution Approach:

1. Rewriting the ODE in standard form: $y'' = \frac{(x - 3) + e^{y'} + (x + 3)^3 - e + y - y^3 - 2xy'}{x^2}.$

2. Numerical or Analytical Solution: This equation is nonlinear due to the terms $y^3$ and $e^{y'}$, making it difficult to solve analytically. Therefore, a numerical approach (such as Euler’s method, Runge-Kutta, or another ODE solver) is required.

3. Applying Initial Conditions: Using $y(1) = 4$ and $y'(1) = 1$, we can integrate numerically from $x = 1$ to $x = 2$.

I will now compute $y(2)$ using a numerical approach. numpy as np from scipy.integrate import solve_ivp

Define the differential equation system

def ode_system(x, Y): y, yp = Y # y = Y[0], yp = Y[1] (y' = yp) dydx = yp dypdx = ((x - 3) + np.exp(yp) + (x + 3)3 - np.exp(1) + y - y3 - 2xyp) / x**2 return [dydx, dypdx]

Initial conditions

x0 = 1 y0 = 4 yp0 = 1 Y0 = [y0, yp0]

Solve the ODE from x = 1 to x = 2

solution = solve_ivp(ode_system, [1, 2], Y0, method='RK45', t_eval=[2])

Extract y(2)

y_2 = solution.y[0, -1] y_2Using numerical integration, we find that $y(2) \approx 5.00$.

Would you like a more detailed explanation of the numerical method used? Here are some related questions:

1. How does the Runge-Kutta method work for solving ODEs?
2. What makes this ODE nonlinear, and how does that affect its solution?
3. Can this equation be solved analytically in any special cases?
4. How does the choice of numerical method affect accuracy?
5. What are common stability issues when solving nonlinear ODEs?

Tip: When dealing with nonlinear differential equations, always check if an analytical solution is feasible before resorting to numerical methods.