function varargout = ode45(ode,tspan,y0,options,varargin)
%ODE45  Solve non-stiff differential equations, medium order method.
%   [TOUT,YOUT] = ODE45(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] = ODE45(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.
%   
%   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
%   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. 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'. ODE15S and ODE23T can solve problems with singular mass matrices.  
%
%   [TOUT,YOUT,TE,YE,IE] = ODE45(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 = ODE45(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 ODE45 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]=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
%       other ODE solvers:    ODE23, ODE113, ODE15S, ODE23S, ODE23T, ODE23TB
%       implicit ODEs:        ODE15I
%       options handling:     ODESET, ODEGET
%       output functions:     ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT
%       evaluating solution:  DEVAL
%       ODE examples:         RIGIDODE, BALLODE, ORBITODE
%       function handles:     FUNCTION_HANDLE
%
%   NOTE:
%     The interpretation of the first input argument of the ODE solvers and
%     some properties available through ODESET have changed in MATLAB 6.0.
%     Although we still support the v5 syntax, any new functionality is
%     available only with the new syntax.  To see the v5 help, type in
%     the command line  
%         more on, type ode45, more off

%   NOTE:
%     This portion describes the v5 syntax of ODE45.
%
%   [T,Y] = ODE45('F',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.  'F' is a string containing the name of an ODE
%   file.  Function F(T,Y) must return a column vector.  Each row in
%   solution array Y corresponds to a time returned in column vector T.  To
%   obtain solutions at specific times T0, T1, ..., TFINAL (all increasing
%   or all decreasing), use TSPAN = [T0 T1 ... TFINAL].
%   
%   [T,Y] = ODE45('F',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).
%   
%   [T,Y] = ODE45('F',TSPAN,Y0,OPTIONS,P1,P2,...) passes the additional
%   parameters P1,P2,... to the ODE file as F(T,Y,FLAG,P1,P2,...) (see
%   ODEFILE).  Use OPTIONS = [] as a place holder if no options are set.
%   
%   It is possible to specify TSPAN, Y0 and OPTIONS in the ODE file (see
%   ODEFILE).  If TSPAN or Y0 is empty, then ODE45 calls the ODE file
%   [TSPAN,Y0,OPTIONS] = F([],[],'init') to obtain any values not supplied
%   in the ODE45 argument list.  Empty arguments at the end of the call list
%   may be omitted, e.g. ODE45('F').
%   
%   ODE45 can solve problems M(t,y)*y' = F(t,y) with a mass matrix M that is
%   nonsingular.  Use ODESET to set Mass to 'M', 'M(t)', or 'M(t,y)' if the
%   ODE file is coded so that F(T,Y,'mass') returns a constant,
%   time-dependent, or time- and state-dependent mass matrix, respectively.
%   The default value of Mass is 'none'. ODE15S and ODE23T can solve problems
%   with singular mass matrices.
%   
%   [T,Y,TE,YE,IE] = ODE45('F',TSPAN,Y0,OPTIONS) with the Events property in
%   OPTIONS set to 'on', solves as above while also locating zero crossings
%   of an event function defined in the ODE file.  The ODE file must be
%   coded so that F(T,Y,'events') returns appropriate information.  See
%   ODEFILE for details.  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.
%   
%   See also ODEFILE

%   ODE45 is an implementation of the explicit Runge-Kutta (4,5) pair of
%   Dormand and Prince called variously RK5(4)7FM, DOPRI5, DP(4,5) and DP54.
%   It uses a "free" interpolant of order 4 communicated privately by
%   Dormand and Prince.  Local extrapolation is done.

%   Details are to be found in The MATLAB ODE Suite, L. F. Shampine and
%   M. W. Reichelt, SIAM Journal on Scientific Computing, 18-1, 1997.

%   Mark W. Reichelt and Lawrence F. Shampine, 6-14-94
%   Copyright 1984-2009 The MathWorks, Inc.
%   $Revision: 5.74.4.10 $  $Date: 2009/04/21 03:24:15 $

solver_name = 'ode45';

% Check inputs
if nargin < 4
  options = [];
  if nargin < 3
    y0 = [];
    if nargin < 2
      tspan = [];
      if nargin < 1
        error('MATLAB:ode45:NotEnoughInputs',...
              'Not enough input arguments.  See ODE45.');
      end  
    end
  end
end

% Stats
nsteps  = 0;
nfailed = 0;
nfevals = 0;

% Output
FcnHandlesUsed  = isa(ode,'function_handle');
output_sol = (FcnHandlesUsed && (nargout==1));      % sol = odeXX(...)
output_ty  = (~output_sol && (nargout > 0));  % [t,y,...] = odeXX(...)
% There might be no output requested...

sol = []; f3d = [];
if output_sol
  sol.solver = solver_name;
  sol.extdata.odefun = ode;
  sol.extdata.options = options;                       
  sol.extdata.varargin = varargin;  
end  

% Handle solver arguments
[neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn, ...
options, threshold, rtol, normcontrol, normy, hmax, htry, htspan, dataType] = ...
    odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
nfevals = nfevals + 1;

% Handle the output
if nargout > 0
  outputFcn = odeget(options,'OutputFcn',[],'fast');
else
  outputFcn = odeget(options,'OutputFcn',@odeplot,'fast');
end
outputArgs = {};      
if isempty(outputFcn)
  haveOutputFcn = false;
else
  haveOutputFcn = true;
  outputs = odeget(options,'OutputSel',1:neq,'fast');
  if isa(outputFcn,'function_handle')  
    % With MATLAB 6 syntax pass additional input arguments to outputFcn.
    outputArgs = varargin;
  end  
end
refine = max(1,odeget(options,'Refine',4,'fast'));
if ntspan > 2
  outputAt = 'RequestedPoints';         % output only at tspan points
elseif refine <= 1
  outputAt = 'SolverSteps';             % computed points, no refinement
else
  outputAt = 'RefinedSteps';            % computed points, with refinement
  S = (1:refine-1) / refine;
end
printstats = strcmp(odeget(options,'Stats','off','fast'),'on');

% Handle the event function
[haveEventFcn,eventFcn,eventArgs,valt,teout,yeout,ieout] = ...
    odeevents(FcnHandlesUsed,odeFcn,t0,y0,options,varargin);

% Handle the mass matrix
[Mtype, M, Mfun] =  odemass(FcnHandlesUsed,odeFcn,t0,y0,options,varargin);
if Mtype > 0  % non-trivial mass matrix  
  Msingular = odeget(options,'MassSingular','no','fast');
  if strcmp(Msingular,'maybe')
    warning('MATLAB:ode45:MassSingularAssumedNo',['ODE45 assumes ' ...
              'MassSingular is ''no''. See ODE15S or ODE23T.']);
  elseif strcmp(Msingular,'yes')
    error('MATLAB:ode45:MassSingularYes',...
          ['MassSingular cannot be ''yes'' for this solver.  See ODE15S '...
           ' or ODE23T.']);
  end
  % Incorporate the mass matrix into odeFcn and odeArgs.
  [odeFcn,odeArgs] = odemassexplicit(FcnHandlesUsed,Mtype,odeFcn,odeArgs,Mfun,M);
  f0 = feval(odeFcn,t0,y0,odeArgs{:});
  nfevals = nfevals + 1;
end

% Non-negative solution components
idxNonNegative = odeget(options,'NonNegative',[],'fast');
nonNegative = ~isempty(idxNonNegative);
if nonNegative  % modify the derivative function
  [odeFcn,thresholdNonNegative] = odenonnegative(odeFcn,y0,threshold,idxNonNegative);
  f0 = feval(odeFcn,t0,y0,odeArgs{:});
  nfevals = nfevals + 1;
end

t = t0;
y = y0;

% Allocate memory if we're generating output.
nout = 0;
tout = []; yout = [];
if nargout > 0
  if output_sol
    chunk = min(max(100,50*refine), refine+floor((2^11)/neq));      
    tout = zeros(1,chunk,dataType);
    yout = zeros(neq,chunk,dataType);
    f3d  = zeros(neq,7,chunk,dataType);
  else      
    if ntspan > 2                         % output only at tspan points
      tout = zeros(1,ntspan,dataType);
      yout = zeros(neq,ntspan,dataType);
    else                                  % alloc in chunks
      chunk = min(max(100,50*refine), refine+floor((2^13)/neq));
      tout = zeros(1,chunk,dataType);
      yout = zeros(neq,chunk,dataType);
    end
  end  
  nout = 1;
  tout(nout) = t;
  yout(:,nout) = y;  
end

% Initialize method parameters.
pow = 1/5;
A = [1/5, 3/10, 4/5, 8/9, 1, 1];
B = [
    1/5         3/40    44/45   19372/6561      9017/3168       35/384
    0           9/40    -56/15  -25360/2187     -355/33         0
    0           0       32/9    64448/6561      46732/5247      500/1113
    0           0       0       -212/729        49/176          125/192
    0           0       0       0               -5103/18656     -2187/6784
    0           0       0       0               0               11/84
    0           0       0       0               0               0
    ];
E = [71/57600; 0; -71/16695; 71/1920; -17253/339200; 22/525; -1/40];
f = zeros(neq,7,dataType);
hmin = 16*eps(t);
if isempty(htry)
  % Compute an initial step size h using y'(t).
  absh = min(hmax, htspan);
  if normcontrol
    rh = (norm(f0) / max(normy,threshold)) / (0.8 * rtol^pow);
  else
    rh = norm(f0 ./ max(abs(y),threshold),inf) / (0.8 * rtol^pow);
  end
  if absh * rh > 1
    absh = 1 / rh;
  end
  absh = max(absh, hmin);
else
  absh = min(hmax, max(hmin, htry));
end
f(:,1) = f0;

% Initialize the output function.
if haveOutputFcn
  feval(outputFcn,[t tfinal],y(outputs),'init',outputArgs{:});
end

% THE MAIN LOOP

done = false;
while ~done
  
  % By default, hmin is a small number such that t+hmin is only slightly
  % different than t.  It might be 0 if t is 0.
  hmin = 16*eps(t);
  absh = min(hmax, max(hmin, absh));    % couldn't limit absh until new hmin
  h = tdir * absh;
  
  % Stretch the step if within 10% of tfinal-t.
  if 1.1*absh >= abs(tfinal - t)
    h = tfinal - t;
    absh = abs(h);
    done = true;
  end
  
  % LOOP FOR ADVANCING ONE STEP.
  nofailed = true;                      % no failed attempts
  while true
    hA = h * A;
    hB = h * B;
    f(:,2) = feval(odeFcn,t+hA(1),y+f*hB(:,1),odeArgs{:});
    f(:,3) = feval(odeFcn,t+hA(2),y+f*hB(:,2),odeArgs{:});
    f(:,4) = feval(odeFcn,t+hA(3),y+f*hB(:,3),odeArgs{:});
    f(:,5) = feval(odeFcn,t+hA(4),y+f*hB(:,4),odeArgs{:});
    f(:,6) = feval(odeFcn,t+hA(5),y+f*hB(:,5),odeArgs{:});

    tnew = t + hA(6);
    if done
      tnew = tfinal;   % Hit end point exactly.
    end
    h = tnew - t;      % Purify h.     
   
    ynew = y + f*hB(:,6);
    f(:,7) = feval(odeFcn,tnew,ynew,odeArgs{:});
    nfevals = nfevals + 6;              
   
    % Estimate the error.
    NNrejectStep = false;
    if normcontrol
      normynew = norm(ynew);
      errwt = max(max(normy,normynew),threshold);
      err = absh * (norm(f * E) / errwt);
      if nonNegative && (err <= rtol) && any(ynew(idxNonNegative)<0)
        errNN = norm( max(0,-ynew(idxNonNegative)) ) / errwt ;
        if errNN > rtol
          err = errNN;
          NNrejectStep = true;
        end
      end      
    else
      err = absh * norm((f * E) ./ max(max(abs(y),abs(ynew)),threshold),inf);
      if nonNegative && (err <= rtol) && any(ynew(idxNonNegative)<0)
        errNN = norm( max(0,-ynew(idxNonNegative)) ./ thresholdNonNegative, inf);      
        if errNN > rtol
          err = errNN;
          NNrejectStep = true;
        end
      end            
    end
   
    % Accept the solution only if the weighted error is no more than the
    % tolerance rtol.  Estimate an h that will yield an error of rtol on
    % the next step or the next try at taking this step, as the case may be,
    % and use 0.8 of this value to avoid failures.
    if err > rtol                       % Failed step
      nfailed = nfailed + 1;            
      if absh <= hmin
        warning('MATLAB:ode45:IntegrationTolNotMet',['Failure at t=%e.  ' ...
                  'Unable to meet integration tolerances without reducing ' ...
                  'the step size below the smallest value allowed (%e) ' ...
                  'at time t.'],t,hmin);
      
        solver_output = odefinalize(solver_name, sol,...
                                    outputFcn, outputArgs,...
                                    printstats, [nsteps, nfailed, nfevals],...
                                    nout, tout, yout,...
                                    haveEventFcn, teout, yeout, ieout,...
                                    {f3d,idxNonNegative});
        if nargout > 0
          varargout = solver_output;
        end  
        return;
      end
      
      if nofailed
        nofailed = false;
        if NNrejectStep
          absh = max(hmin, 0.5*absh);
        else
          absh = max(hmin, absh * max(0.1, 0.8*(rtol/err)^pow));
        end
      else
        absh = max(hmin, 0.5 * absh);
      end
      h = tdir * absh;
      done = false;
      
    else                                % Successful step

      NNreset_f7 = false;
      if nonNegative && any(ynew(idxNonNegative)<0)
        ynew(idxNonNegative) = max(ynew(idxNonNegative),0);
        if normcontrol
          normynew = norm(ynew);
        end
        NNreset_f7 = true;
      end  
                  
      break;
      
    end
  end
  nsteps = nsteps + 1;                  
  
  if haveEventFcn
    [te,ye,ie,valt,stop] = ...
        odezero(@ntrp45,eventFcn,eventArgs,valt,t,y,tnew,ynew,t0,h,f,idxNonNegative);
    if ~isempty(te)
      if output_sol || (nargout > 2)
        teout = [teout, te];
        yeout = [yeout, ye];
        ieout = [ieout, ie];
      end
      if stop               % Stop on a terminal event.               
        % Adjust the interpolation data to [t te(end)].   
        
        % Update the derivatives using the interpolating polynomial.
        taux = t + (te(end) - t)*A;        
        [~,f(:,2:7)] = ntrp45(taux,t,y,[],[],h,f,idxNonNegative);        
        
        tnew = te(end);
        ynew = ye(:,end);
        h = tnew - t;
        done = true;
      end
    end
  end

  if output_sol
    nout = nout + 1;
    if nout > length(tout)
      tout = [tout, zeros(1,chunk,dataType)];  % requires chunk >= refine
      yout = [yout, zeros(neq,chunk,dataType)];
      f3d  = cat(3,f3d,zeros(neq,7,chunk,dataType));
    end
    tout(nout) = tnew;
    yout(:,nout) = ynew;
    f3d(:,:,nout) = f;
  end  
   
  if output_ty || haveOutputFcn
    switch outputAt
     case 'SolverSteps'        % computed points, no refinement
      nout_new = 1;
      tout_new = tnew;
      yout_new = ynew;
     case 'RefinedSteps'       % computed points, with refinement
      tref = t + (tnew-t)*S;
      nout_new = refine;
      tout_new = [tref, tnew];
      yout_new = [ntrp45(tref,t,y,[],[],h,f,idxNonNegative), ynew];
     case 'RequestedPoints'    % output only at tspan points
      nout_new =  0;
      tout_new = [];
      yout_new = [];
      while next <= ntspan  
        if tdir * (tnew - tspan(next)) < 0
          if haveEventFcn && stop     % output tstop,ystop
            nout_new = nout_new + 1;
            tout_new = [tout_new, tnew];
            yout_new = [yout_new, ynew];            
          end
          break;
        end
        nout_new = nout_new + 1;              
        tout_new = [tout_new, tspan(next)];
        if tspan(next) == tnew
          yout_new = [yout_new, ynew];            
        else  
          yout_new = [yout_new, ntrp45(tspan(next),t,y,[],[],h,f,idxNonNegative)];
        end  
        next = next + 1;
      end
    end
   
    if nout_new > 0
      if output_ty
        oldnout = nout;
        nout = nout + nout_new;
        if nout > length(tout)
          tout = [tout, zeros(1,chunk,dataType)];  % requires chunk >= refine
          yout = [yout, zeros(neq,chunk,dataType)];
        end
        idx = oldnout+1:nout;        
        tout(idx) = tout_new;
        yout(:,idx) = yout_new;
      end
      if haveOutputFcn
        stop = feval(outputFcn,tout_new,yout_new(outputs,:),'',outputArgs{:});
        if stop
          done = true;
        end  
      end     
    end  
  end
  
  if done
    break
  end

  % If there were no failures compute a new h.
  if nofailed
    % Note that absh may shrink by 0.8, and that err may be 0.
    temp = 1.25*(err/rtol)^pow;
    if temp > 0.2
      absh = absh / temp;
    else
      absh = 5.0*absh;
    end
  end
  
  % Advance the integration one step.
  t = tnew;
  y = ynew;
  if normcontrol
    normy = normynew;
  end
  if NNreset_f7
    % Used f7 for unperturbed solution to interpolate.  
    % Now reset f7 to move along constraint.
    f(:,7) = feval(odeFcn,tnew,ynew,odeArgs{:});
    nfevals = nfevals + 1;
  end
  f(:,1) = f(:,7);  % Already have f(tnew,ynew)
  
end

solver_output = odefinalize(solver_name, sol,...
                            outputFcn, outputArgs,...
                            printstats, [nsteps, nfailed, nfevals],...
                            nout, tout, yout,...
                            haveEventFcn, teout, yeout, ieout,...
                            {f3d,idxNonNegative});
if nargout > 0
  varargout = solver_output;
end  

function jie_odes
clear all;clc
format long
tspan=[0 1.524];
y0=[0,661.4194946];
[t y]=ode45(@obj,tspan,y0);
t
l=y(:,1)
x=y(:,2)
figure(1)
subplot(2,1,1)
plot(t,y(:,1),'b-');
subplot(2,1,2);
plot(t,y(:,2),'r--');


function f=obj(t,y)
s=0.883338769073639;
z=1;
l=y(1);
x=y(2);

f(1)=(-135.4376*(-26.29682+0.03645*t)*(58325*(1-x)-(58325*x*s)^2/(-0.03753273+0.00173326*t))/s*z);
f(2)=((81.3+42.97*s+(25.75257+10.317*s)*t+1/2*(162.886-2.04899999999998*s)*t^2+1/3*(-42.4392-3.8055*s)*t^3)/(1331.847424*(1+x))/((0.23743+25.75257*x+10.317*x*s)+...
    (32.28+160.837*x-2.04899999999999*x*s)*t+(-11.5698-46.2447*x-3.8055*x*s)*t^2)*f(1));
f=f';