This problem involves solving a boundary value problem for the beam deflection equation using the shooting method. The deflection of a beam subjected to a distributed load is described by the equation:

$EI \frac{d^2y}{dx^2} = \frac{1}{2} wLx - \frac{1}{2} wx^2$

Where:

• $E$ is the Young's modulus (201 kN/m²),
• $I$ is the moment of inertia (4.5730 kg.m²),
• $w$ is the uniform load (15.0 kN/m),
• $L$ is the length of the beam (10 m),
• $y(0) = 0$ and $y(10) = 0$ are the boundary conditions.

Step 1: Reformulate the equation

To apply the shooting method, we first rewrite the second-order differential equation as a system of first-order ODEs. We introduce a new variable $v(x)$ such that:

$v(x) = \frac{dy}{dx}$

Thus, the system becomes:

• $\frac{dy}{dx} = v(x)$,
• $\frac{dv}{dx} = \frac{1}{2} \frac{wLx - wx^2}{EI}$.

This gives us two equations:

1. $\frac{dy}{dx} = v(x)$,
2. $\frac{dv}{dx} = \frac{1}{2} \frac{wLx - wx^2}{EI}$.

Step 2: Boundary conditions

The boundary conditions are:

• $y(0) = 0$,
• $y(10) = 0$.

For the shooting method, we need to guess an initial value for $v(0)$, solve the system, and adjust the guess until we get $y(10) = 0$.

Step 3: Euler method

To numerically solve the system, we use the Euler method, which approximates the solution using the following iterative scheme:

$y_{n+1} = y_n + h v_n$ $v_{n+1} = v_n + h \frac{1}{2EI} (wLx_n - wx_n^2)$

Where $h = 2$ m is the step size, and $x_n$ is the point at each iteration.

Step 4: Applying the RK2 method (Runge-Kutta 2nd order)

We also need to use a second-order Runge-Kutta method (RK2) for better accuracy. The RK2 method updates the values of $y$ and $v$ as follows:

1. Intermediate slopes: $k_1 = h f(x_n, y_n, v_n)$ $k_2 = h f(x_n + h, y_n + k_1, v_n + k_1)$

2. Update: $y_{n+1} = y_n + \frac{1}{2}(k_1 + k_2)$ $v_{n+1} = v_n + \frac{1}{2}(k_1 + k_2)$

This allows for better approximation than the simple Euler method.

Step 5: Iteration and shooting method

Now, using an initial guess for $v(0)$, you would:

1. Apply Euler or RK2 method to the system,
2. Check the value of $y(10)$,
3. Adjust the guess for $v(0)$ if necessary, and repeat the process until $y(10)$ is close to 0.

Let me know if you need further clarification on any step, or if you would like me to solve for specific values or iterations based on your setup!