How to solve a 13 set of equations and evaluate their change over time?

ODE15S Solve stiff differential equations and DAEs, variable order method. [TOUT,YOUT] = ODE15S(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates the system of differential equations y' = f(t,y) from time T0 to TFINAL with initial conditions Y0. 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). Each row in the solution array YOUT corresponds to a time returned in the column vector TOUT. To obtain solutions at specific times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN = [T0 T1 ... TFINAL]. [TOUT,YOUT] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default integration properties replaced by values in OPTIONS, an argument created with the ODESET function. See ODESET for details. Commonly used options are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector of absolute error tolerances 'AbsTol' (all components 1e-6 by default). If certain components of the solution must be non-negative, use ODESET to set the 'NonNegative' property to the indices of these components. The 'NonNegative' property is ignored for problems where there is a mass matrix. The Jacobian matrix df/dy is critical to reliability and efficiency. Use ODESET to set 'Jacobian' to a function handle FJAC if FJAC(T,Y) returns the Jacobian df/dy or to the matrix df/dy if the Jacobian is constant. If the 'Jacobian' option is not set (the default), df/dy is approximated by finite differences. Set 'Vectorized' 'on' if the ODE function is coded so that ODEFUN(T,[Y1 Y2 ...]) returns [ODEFUN(T,Y1) ODEFUN(T,Y2) ...]. If df/dy is a sparse matrix, set 'JPattern' to the sparsity pattern of df/dy, i.e., a sparse matrix S with S(i,j) = 1 if component i of f(t,y) depends on component j of y, and 0 otherwise. ODE15S can solve problems M(t,y)*y' = f(t,y) with mass matrix M(t,y). Use ODESET to set the 'Mass' property to a function handle MASS if MASS(T,Y) returns the value of the mass matrix. If the mass matrix is constant, the matrix can be used as the value of the 'Mass' option. Problems with state-dependent mass matrices are more difficult. If the mass matrix does not depend on the state variable Y and the function MASS is to be called with one input argument T, set 'MStateDependence' to 'none'. If the mass matrix depends weakly on Y, set 'MStateDependence' to 'weak' (the default) and otherwise, to 'strong'. In either case the function MASS is to be called with the two arguments (T,Y). If there are many differential equations, it is important to exploit sparsity: Return a sparse M(t,y). Either supply the sparsity pattern of df/dy using the 'JPattern' property or a sparse df/dy using the Jacobian property. For strongly state-dependent M(t,y), set 'MvPattern' to a sparse matrix S with S(i,j) = 1 if for any k, the (i,k) component of M(t,y) depends on component j of y, and 0 otherwise. If the mass matrix is non-singular, the solution of the problem is straightforward. See examples FEM1ODE, FEM2ODE, BATONODE, or BURGERSODE. If M(t0,y0) is singular, the problem is a differential- algebraic equation (DAE). ODE15S solves DAEs of index 1. DAEs have solutions only when y0 is consistent, i.e., there is a yp0 such that M(t0,y0)*yp0 = f(t0,y0). Use ODESET to set 'MassSingular' to 'yes', 'no', or 'maybe'. The default of 'maybe' causes ODE15S to test whether M(t0,y0) is singular. You can provide yp0 as the value of the 'InitialSlope' property. The default is the zero vector. If y0 and yp0 are not consistent, ODE15S treats them as guesses, tries to compute consistent values close to the guesses, and then goes on to solve the problem. See examples HB1DAE or AMP1DAE. [TOUT,YOUT,TE,YE,IE] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events' property in OPTIONS set to a function handle EVENTS, solves as above while also finding where functions of (T,Y), called event functions, are zero. For each function you specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. These are the three column vectors returned by EVENTS: [VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function: VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration is to terminate at a zero of this event function and 0 otherwise. DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only zeros where the event function is increasing, and -1 if only zeros where the event function is decreasing. Output TE is a column vector of times at which events occur. Rows of YE are the corresponding solutions, and indices in vector IE specify which event occurred. SOL = ODE15S(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be used with DEVAL to evaluate the solution or its first derivative at any point between T0 and TFINAL. The steps chosen by ODE15S are returned in a row vector SOL.x. For each I, the column SOL.y(:,I) contains the solution at SOL.x(I). If events were detected, SOL.xe is a row vector of points at which events occurred. Columns of SOL.ye are the corresponding solutions, and indices in vector SOL.ie specify which event occurred. Example [t,y]=ode15s(@vdp1000,[0 3000],[2 0]); plot(t,y(:,1)); solves the system y' = vdp1000(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. See also ODE23S, ODE23T, ODE23TB, ODE45, ODE23, ODE113, ODE15I, ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL, ODEEXAMPLES, VDPODE, BRUSSODE, HB1DAE, FUNCTION_HANDLE. Documentation for ode15s doc ode15s

ODE23T Solve moderately stiff ODEs and DAEs, trapezoidal rule. [TOUT,YOUT] = ODE23T(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates the system of differential equations y' = f(t,y) from time T0 to TFINAL with initial conditions Y0. 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). Each row in the solution array YOUT corresponds to a time returned in the column vector TOUT. To obtain solutions at specific times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN = [T0 T1 ... TFINAL]. [TOUT,YOUT] = ODE23T(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default integration parameters replaced by values in OPTIONS, an argument created with the ODESET function. See ODESET for details. Commonly used options are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector of absolute error tolerances 'AbsTol' (all components 1e-6 by default). If certain components of the solution must be non-negative, use ODESET to set the 'NonNegative' property to the indices of these components. The 'NonNegative' property is ignored for problems where there is a mass matrix. The Jacobian matrix df/dy is critical to reliability and efficiency. Use ODESET to set 'Jacobian' to a function handle FJAC if FJAC(T,Y) returns the Jacobian df/dy or to the matrix df/dy if the Jacobian is constant. If the 'Jacobian' option is not set (the default), df/dy is approximated by finite differences. Set 'Vectorized' 'on' if the ODE function is coded so that ODEFUN(T,[Y1 Y2 ...]) returns [ODEFUN(T,Y1) ODEFUN(T,Y2) ...]. If df/dy is a sparse matrix, set 'JPattern' to the sparsity pattern of df/dy, i.e., a sparse matrix S with S(i,j) = 1 if component i of f(t,y) depends on component j of y, and 0 otherwise. ODE23T can solve problems M(t,y)*y' = f(t,y) with mass matrix M(t,y). Use ODESET to set the 'Mass' property to a function handle MASS if MASS(T,Y) returns the value of the mass matrix. If the mass matrix is constant, the matrix can be used as the value of the 'Mass' option. Problems with state-dependent mass matrices are more difficult. If the mass matrix does not depend on the state variable Y and the function MASS is to be called with one input argument T, set 'MStateDependence' to 'none'. If the mass matrix depends weakly on Y, set 'MStateDependence' to 'weak' (the default) and otherwise, to 'strong'. In either case the function MASS is to be called with the two arguments (T,Y). If there are many differential equations, it is important to exploit sparsity: Return a sparse M(t,y). Either supply the sparsity pattern of df/dy using the 'JPattern' property or a sparse df/dy using the Jacobian property. For strongly state-dependent M(t,y), set 'MvPattern' to a sparse matrix S with S(i,j) = 1 if for any k, the (i,k) component of M(t,y) depends on component j of y, and 0 otherwise. If the mass matrix is non-singular, the solution of the problem is straightforward. See examples FEM1ODE, FEM2ODE, BATONODE, or BURGERSODE. If M(t0,y0) is singular, the problem is a differential- algebraic equation (DAE). ODE23T solves DAEs of index 1. DAEs have solutions only when y0 is consistent, i.e., there is a yp0 such that M(t0,y0)*yp0 = f(t0,y0). Use ODESET to set 'MassSingular' to 'yes', 'no', or 'maybe'. The default of 'maybe' causes ODE23T to test whether M(t0,y0) is singular. You can provide yp0 as the value of the 'InitialSlope' property. The default is the zero vector. If y0 and yp0 are not consistent, ODE23T treats them as guesses, tries to compute consistent values close to the guesses, and then goes on to solve the problem. See examples HB1DAE or AMP1DAE. [TOUT,YOUT,TE,YE,IE] = ODE23T(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events' property in OPTIONS set to a function handle EVENTS, solves as above while also finding where functions of (T,Y), called event functions, are zero. For each function you specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. These are the three column vectors returned by EVENTS: [VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function: VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration is to terminate at a zero of this event function and 0 otherwise. DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only zeros where the event function is increasing, and -1 if only zeros where the event function is decreasing. Output TE is a column vector of times at which events occur. Rows of YE are the corresponding solutions, and indices in vector IE specify which event occurred. SOL = ODE23T(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be used with DEVAL to evaluate the solution or its first derivative at any point between T0 and TFINAL. The steps chosen by ODE23T are returned in a row vector SOL.x. For each I, the column SOL.y(:,I) contains the solution at SOL.x(I). If events were detected, SOL.xe is a row vector of points at which events occurred. Columns of SOL.ye are the corresponding solutions, and indices in vector SOL.ie specify which event occurred. Example [t,y]=ode23t(@vdp1000,[0 3000],[2 0]); plot(t,y(:,1)); solves the system y' = vdp1000(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. See also ODE15S, ODE23S, ODE23TB, ODE45, ODE23, ODE113, ODE15I, ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL, ODEEXAMPLES, VDPODE, BRUSSODE, HB1DAE, FUNCTION_HANDLE. Documentation for ode23t doc ode23t