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Resolver la ecuación encerrada en rojo por el metodo Runge Kutta

To solve the differential equation enclosed in red using the Runge-Kutta method, we first need to identify the form of the equation. The equation is a second-order linear differential equation given by:

\[ V_i(t) = LC \frac{d^2 V_c(t)}{dt^2} + RC \frac{d V_c(t)}{dt} + V_c(t) \]

Given the values:
- \( L = 2 \text{ mH} = 2 \times 10^{-3} \text{ H} \)
- \( C = 100 \mu\text{F} = 100 \times 10^{-6} \text{ F} \)
- \( R = 1 \text{ k}\Omega = 1 \times 10^3 \Omega \)
- \( V_i(t) = 12 \text{ V} \)

The equation can be rewritten as:
\[ 12 = (2 \times 10^{-3} \times 100 \times 10^{-6}) \frac{d^2 V_c(t)}{dt^2} + (1 \times 10^3 \times 100 \times 10^{-6}) \frac{d V_c(t)}{dt} + V_c(t) \]

Simplifying the constants:
\[ 12 = 2 \times 10^{-8} \frac{d^2 V_c(t)}{dt^2} + 0.1 \frac{d V_c(t)}{dt} + V_c(t) \]

To apply the Runge-Kutta method, we need to convert this second-order differential equation into a system of first-order differential equations. Let's define:
\[ y_1 = V_c(t) \]
\[ y_2 = \frac{d V_c(t)}{dt} \]

Thus,
\[ \frac{d y_1}{dt} = y_2 \]
\[ \frac{d y_2}{dt} = \frac{1}{2 \times 10^{-8}} \left( 12 - 0.1 y_2 - y_1 \right) \]

Let's solve this system using the Runge-Kutta method (4th order). The equations for \( y_1 \) and \( y_2 \) are:
\[ y_1' = y_2 \]
\[ y_2' = \frac{1}{2 \times 10^{-8}} (12 - 0.1 y_2 - y_1) \]

Now we can implement the Runge-Kutta method:

### Step-by-Step Solution using Runge-Kutta Method

1. Define initial conditions: \( y_1(0) = V_{c0} \) and \( y_2(0) = \frac{d V_c(0)}{dt} \).

2. Choose a step size \( h \) for the time variable \( t \).

3. Apply the Runge-Kutta update rules for \( y_1 \) and \( y_2 \):

   \[ k_1 = h f(t_n, y_{1n}, y_{2n}) \]
   \[ l_1 = h g(t_n, y_{1n}, y_{2n}) \]

   \[ k_2 = h f(t_n + \frac{h}{2}, y_{1n} + \frac{k_1}{2}, y_{2n} + \frac{l_1}{2}) \]
   \[ l_2 = h g(t_n + \frac{h}{2}, y_{1n} + \frac{k_1}{2}, y_{2n} + \frac{l_1}{2}) \]

   \[ k_3 = h f(t_n + \frac{h}{2}, y_{1n} + \frac{k_2}{2}, y_{2n} + \frac{l_2}{2}) \]
   \[ l_3 = h g(t_n + \frac{h}{2}, y_{1n} + \frac{k_2}{2}, y_{2n} + \frac{l_2}{2}) \]

   \[ k_4 = h f(t_n + h, y_{1n} + k_3, y_{2n} + l_3) \]
   \[ l_4 = h g(t_n + h, y_{1n} + k_3, y_{2n} + l_3) \]

   Where:
   \[ f(t, y_1, y_2) = y_2 \]
   \[ g(t, y_1, y_2) = \frac{1}{2 \times 10^{-8}} (12 - 0.1 y_2 - y_1) \]

4. Update the values:
   \[ y_{1(n+1)} = y_{1n} + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6} \]
   \[ y_{2(n+1)} = y_{2n} + \frac{l_1 + 2l_2 + 2l_3 + l_4}{6} \]

5. Repeat steps 3-4 for the desired number of steps or until the desired time is reached.

Let me calculate the first few steps of this process.

Learn how to solve a second-order differential equation using the Runge-Kutta method. This problem involves applying numerical integration techniques to solve for voltage over a capacitor in an electrical circuit. Suitable for advanced college-level students studying differential equations and numerical methods.

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