Question

Mole olutiins aAe

Answer 1

%%Matlab code for ode solution and Phase potrait clear all close all %function to be solved x-ini-o ; %initial time x, end-20; %ending time Plotting Direction field figure(1) dirrfield (f,-5:.5:5,-5:.5:5) hold on %plot for different initial conditions y0-[-1-2 1-41: x0 [0 0 0 0] for i-1:4 [ts,ys]ode45(f, [x ini,x_end],v0); plot (ts,ys, 'b', 'Linewidth',2) end hold off title( 'phase potrait plot') xlabel(X ylabel('y' %%Matlab function for direction filed function dirrfield (f,tval,yval) % dirfield ( f , t1:dt:t2, yldy : y2) plot direction field for first order ODE y' f(t,y) using t-values from tl to t2 with spacing of dt using y-values from yl to t2 with spacing of dy 8 f is an function, or an inline function, or the name of an m-file with quotes. % Example: y' --y 8 Show direction field for t in [-1,3, y in [-2,2, use spacing of .2 for both t and y: -у"2+t dirfield(f, -1:.2:3, -2:.2:2) f @(t,y) [tm, ym]-meshgrid (tval,yval); dt = tval (2) -tval (1); dy = yval ( 2 ) -yval ( 1 ) ; fv - vectorize(f); if isa(f, function handle')

fv - eval (fv) end yp-feval (fv, tm, ym); s-1./max (1/dt, abs (yp)./dy) *0.75; h-ishold; quiver(tval,yval,s,s.*yp,0, &quiver ( tval , yval ,-s ,-s. * yp , 0 , ' r " ) ; ifh hold on; hold on else hold off end axis ([tval (1)-dt/2,tval (end) +dt/2,yval 1)- dy/2,yval(end)+dy/2]) end phase potrait plot Equilibrium solution can be achieved at y -3 and y 0. At y-3 it is stable but for y 0 it is unstable. Now for initial condition y(0)--1; y(O)--2 and y(0)--4 the solution becomes stable but for y(0)-1 solution become unstable and diverges to infinity

%%Matlab code for ode solution and Phase potrait
clear all
close all
%function to be solved
f = @(x,y) (1/2).*y.*(y+3);

x_ini=0;   %initial time
x_end=20; %ending time
%Plotting Direction field
figure(1)
dirrfield(f,-5:.5:5,-5:.5:5)
hold on
%plot for different initial conditions
y0=[-1 -2 1 -4];
x0=[0 0 0 0];
for i=1:4
x_ini=x0(i); v0=y0(i);
[ts,ys] = ode45(f,[x_ini,x_end],v0); plot(ts,ys,'b','Linewidth',2)
end
hold off
title('phase potrait plot')
xlabel('x')
ylabel('y')

%%Matlab function for direction filed
function dirrfield(f,tval,yval)
% dirfield(f, t1:dt:t2, y1:dy:y2)
%
%   plot direction field for first order ODE y' = f(t,y)
%   using t-values from t1 to t2 with spacing of dt
%   using y-values from y1 to t2 with spacing of dy
%
%   f is an @ function, or an inline function,
%     or the name of an m-file with quotes.
%
% Example: y' = -y^2 + t
%   Show direction field for t in [-1,3], y in [-2,2], use
%   spacing of .2 for both t and y:
%
%   f = @(t,y) -y^2+t
%   dirfield(f, -1:.2:3, -2:.2:2)

    [tm,ym]=meshgrid(tval,yval);
    dt = tval(2) - tval(1);
    dy = yval(2) - yval(1);
    fv = vectorize(f);
    if isa(f,'function_handle')
      fv = eval(fv);
    end
    yp=feval(fv,tm,ym);
    s = 1./max(1/dt,abs(yp)./dy)*0.75;
    h = ishold;
    quiver(tval,yval,s,s.*yp,0,'r'); hold on;
    %quiver(tval,yval,-s,-s.*yp,0,'r');
    if h
      hold on
    else
      hold off
    end
    axis([tval(1)-dt/2,tval(end)+dt/2,yval(1)-dy/2,yval(end)+dy/2])
end

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%