An autonomous system of two first-order differential equations can be written as:

\[ \frac{du}{dt} = f(u, v) \]
\[ \frac{dv}{dt} = g(u, v) \]

A third-order explicit Runge-Kutta scheme for an autonomous system of two first-order equations is:

\[ k_1 = h \cdot f(u_n, v_n) \]
\[ l_1 = h \cdot g(u_n, v_n) \]

\[ k_2 = h \cdot f(u_n + \frac{1}{2}k_1, v_n + \frac{1}{2}l_1) \]
\[ l_2 = h \cdot g(u_n + \frac{1}{2}k_1, v_n + \frac{1}{2}l_1) \]

\[ k_3 = h \cdot f(u_n - k_1 + 2k_2, v_n - l_1 + 2l_2) \]
\[ l_3 = h \cdot g(u_n - k_1 + 2k_2, v_n - l_1 + 2l_2) \]

The updated values are:

\[ u_{n+1} = u_n + \frac{1}{6}(k_1 + 4k_2 + k_3) \]
\[ v_{n+1} = v_n + \frac{1}{6}(l_1 + 4l_2 + l_3) \]

Consider these equations to implement the Runge-Kutta method for solving the system.

The third order explicit Runge-Kutta scheme is a numerical method used to approximate the solutions of an autonomous system of two first order differential equations. It provides a higher level of accuracy in approximating the solutions by evaluating the derivatives at multiple intermediate points within each step of the numerical integration process.

Explanation:

An autonomous system of two first order differential equations represents a system of equations where the derivatives of the unknown functions are expressed in terms of the functions themselves. These types of systems often arise in various fields of science and engineering, such as physics and biology.

The Runge-Kutta method is a numerical method used to approximate the solutions of ordinary differential equations. It is particularly useful for solving autonomous systems of equations. The third order explicit Runge-Kutta scheme is a specific variant of the Runge-Kutta method that provides a higher level of accuracy in approximating the solutions of the system.

The third order explicit Runge-Kutta scheme involves evaluating the derivatives at multiple intermediate points within each step of the numerical integration process. By using this scheme, we can obtain more accurate and reliable solutions for the given autonomous system of two first order equations.

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