RK4 method for solving first order equations

ODE45 Solve non-stiff differential equations, medium order method. [TOUT,YOUT] = ODE45(ODEFUN,TSPAN,Y0) integrates the system of differential equations y' = f(t,y) from time TSPAN(1) to TSPAN(end) with initial conditions Y0. Each row in the solution array YOUT corresponds to a time in the column vector TOUT. * ODEFUN is a function handle. For a scalar T and a vector Y, ODEFUN(T,Y) must return a column vector corresponding to f(t,y). * TSPAN is a two-element vector [T0 TFINAL] or a vector with several time points [T0 T1 ... TFINAL]. If you specify more than two time points, ODE45 returns interpolated solutions at the requested times. * YO is a column vector of initial conditions, one for each equation. [TOUT,YOUT] = ODE45(ODEFUN,TSPAN,Y0,OPTIONS) specifies integration option values in the fields of a structure, OPTIONS. Create the options structure with odeset. [TOUT,YOUT,TE,YE,IE] = ODE45(ODEFUN,TSPAN,Y0,OPTIONS) produces additional outputs for events. An event occurs when a specified function of T and Y is equal to zero. See ODE Event Location for details. SOL = ODE45(...) returns a solution structure instead of numeric vectors. Use SOL as an input to DEVAL to evaluate the solution at specific points. Use it as an input to ODEXTEND to extend the integration interval. ODE45 can solve problems M(t,y)*y' = f(t,y) with mass matrix M that is nonsingular. Use ODESET to set the 'Mass' property to a function handle or the value of the mass matrix. ODE15S and ODE23T can solve problems with singular mass matrices. ODE23, ODE45, ODE78, and ODE89 are all single-step solvers that use explicit Runge-Kutta formulas of different orders to estimate the error in each step. * ODE45 is for general use. * ODE23 is useful for moderately stiff problems. * ODE78 and ODE89 may be more efficient than ODE45 on non-stiff problems that are smooth except possibly for a few isolated discontinuities. * ODE89 may be more efficient than ODE78 on very smooth problems, when integrating over long time intervals, or when tolerances are tight. Example [t,y]=ode45(@vdp1,[0 20],[2 0]); plot(t,y(:,1)); solves the system y' = vdp1(t,y), using the default relative error tolerance 1e-3 and the default absolute tolerance of 1e-6 for each component, and plots the first component of the solution. Class support for inputs TSPAN, Y0, and the result of ODEFUN(T,Y): float: double, single See also ODE23, ODE78, ODE89, ODE113, ODE15S, ODE23S, ODE23T, ODE23TB, ODE15I, ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL, ODEEXAMPLES, FUNCTION_HANDLE. Documentation for ode45 doc ode45 Other uses of ode45 dlarray/ode45