Question

solve in matlab please

Draw the phase line for ay-ay(y2-1)(2+y), α > 0 and then using Matlab Problem 1 : graph the solution for different initial conditions. Once you complete that, double the value of α and see what happens.

Answer 1

Problem1 兄︸ hone plane (M う

clear all close all %function to be solved a-2; yy-linspace (-2.1,1.1) figure(1) plot (yy, f (0,yy)) title('f(y) vs. y plot for a-2') xlabel('y') ylabel('f(y) ') x ini-0: x end-10; %Plotting Direction field figure (2) dirrfield (f, 0:1: 10,-3:.5:3) hold on %plot for different initial velocity for v0--10:1:10 [ xs , ys ] ode45 ( f , [ x-ini , x-end ] , VO ) ; end hold off title( 'phase potrait plot for a-2') plot ( xs , ys, ' b ' , ' Linewidth' , 2 ) xlabel('x) ylabel( yx) grid orn figure (3) xs, ys] -ode45(f, [x ini,1].-5) plot (xs,ys, 'b, 'Linewidth, 2) title( 'y(x) xlabel'x') x plot for a-2 and y(0)=-5.) vs. ylabel'y(x) ') grid on %function to be solved a-4; yy-linspace (-2.1,1.1) x ini-0; x end-10; ePlotting Direction field figure (4) dirrfield(f, 0:1:10,-3:.5:3) hold on %plot for different initial velocity for v0--10:1:10 [ xs , ys) . ode45 ( f , [ x-ini , x-end ] , v0 ) ; end hold off title 'phase potrait plot for a-4') plot ( xs , ys, ' b' , ' Linewidth' , 2)

xlabel ( ·x' ) ylabel'y(x) ') grid on figure (5) (xs,ys] -ode45(f, [x ini,1],-5); plot (xs,ys, 'b, 'Linewidth',2) title( 'Y (x) vs.x plot for a- and y (0)--5) xlabel('x') ylabel('y(x)') grid on %%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 d 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, 一1:.2:3, -2: .2:2) [tm, ym]-meshgrid(tval,yval) dt = tva 1 (2) -tval (1); dyyval(2) -yval(1); fv - vectorize (f); if isa(f, 'function_handle') end yp-feval (£v, tm, ym)i s-1./max (1/dt, abs (yp)./dy) *0.75; quiver (tval,yval,s,s.*yp,0, 'r'; hold on; quiver (tval,yval,-s,-s.*yp, 0, r' ifh hold on else hold off axis ([tval (1)-dt/2,tval end)+dt/2,yval (1)-dy/2,yval (end) +dy/2]) 용욤욤응%号号%%号号%%号号% End of Code 응응용용융응용욤응응용욤응응용욤 Warning: Failure at t=1.415787e-02. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (2.775558e-17) at time t.

ttv) vs. y plot for a-2 1.5 0.5 0.5 1.5 2.5 1.5 0.5 0.5 1.5 phase potrait plot for a 2 0 234 5 678910

ytx) vs. x plot for a-2 and yo)--5 -2.5 -3.5 -4.5 -5 .1 0 03 04 05 0608 0.9 phase potrait plot for a 4 0 234 5 678910

ytx) vs. x plot for a 4 and ylo)--5 -2 -2.5 .3 -3.5 -4.5 -5 .1 0 03 04 05 0608 0.9 Published with MATLAB R2018a

clear all
close all
%function to be solved
a=2;
f = @(x,y) 2.*y.*(y.^2-1).*(2+y);
yy=linspace(-2.1,1.1);
figure(1)
plot(yy,f(0,yy))
title('f(y) vs. y plot for a=2')
xlabel('y')
ylabel('f(y)')

x_ini=0;
x_end=10;
%Plotting Direction field
figure(2)
dirrfield(f,0:1:10,-3:.5:3)
hold on
%plot for different initial velocity
for v0=-10:1:10
[xs,ys] = ode45(f,[x_ini,x_end],v0); plot(xs,ys,'b','Linewidth',2)
end
hold off
title('phase potrait plot for a=2')
xlabel('x')
ylabel('y(x)')
grid on

figure(3)
[xs,ys] = ode45(f,[x_ini,1],-5); plot(xs,ys,'b','Linewidth',2)
title('y(x) vs. x plot for a=2 and y(0)=-5')
xlabel('x')
ylabel('y(x)')
grid on

%function to be solved
a=4;
f = @(x,y) 4.*y.*(y.^2-1).*(2+y);
yy=linspace(-2.1,1.1);

x_ini=0;
x_end=10;
%Plotting Direction field
figure(4)
dirrfield(f,0:1:10,-3:.5:3)
hold on
%plot for different initial velocity
for v0=-10:1:10
[xs,ys] = ode45(f,[x_ini,x_end],v0); plot(xs,ys,'b','Linewidth',2)
end
hold off
title('phase potrait plot for a=4')
xlabel('x')
ylabel('y(x)')
grid on

figure(5)
[xs,ys] = ode45(f,[x_ini,1],-5); plot(xs,ys,'b','Linewidth',2)
title('y(x) vs. x plot for a=4 and y(0)=-5')
xlabel('x')
ylabel('y(x)')
grid on

%%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 d
%   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 %%%%%%%%%%%%%%%%