We can solve the coupled ODE system by using ‘ode45’ in MATLAB. This function requires arguments as first-order ODE equations, time, and initial conditions. By looking into your equation, the state vector of this system will be [x, y, y_dot], where y_dot is the derivative of y with respect to t.
So you need to rewrite both equations in a form of a column vector consisting of first-order ODEs. The LHS of column vector should look like [x_dot, y_dot, y_ddot], where x_dot and y_dot are derivative of x and y with respect to t respectively, and y_ddot is derivative of y with respect to t.
Please refer to the below link for the examples on ‘ode45’.