Question

MATLAB HELP 3. (a) In one window, graph four different solutions to y 00 + 10y 0 + y = sin t by using different initial conditions. (Be sure that all four graphs are clearly visible in the window.) (b) Describe the apparent behavior of the solutions as t → ∞.

4. (a) Graph solutions to y 00 + a y = sin 3t, y(0) = 1, y 0 (0) = 1 for each of the values a = 9.5, 9.25, 9.125, and 9. (b) Which equation (if any) shows resonance? (Hint: You may need to change the range for x to see the full behavior of each solution.)

(a) In one window, graph four different solutions to y" + 10y' + y = sin t by using different initial conditions. (Be sure that all four graphs are clearly visible in the window.) 3. (b) Describe the apparent behavior of the solutions as t -» oo. (a) Graph solutions to y" + ay = sin3t, y (0) = 1, y'(0) = 1 for each of the values a = 9.5, 9.25, 9.125, and 9 4. (b) Which equation (if any) shows resonance? (Hint: You may need to change the range for x to see the full behavior of each solution.)

Answer 1

Matlab code for solving ode clear all close all Answering question 3 %1st Initial conditions for ode y0 1 01 minimum and maximum time span tspan-[0 50 ] Solution of ODES using ode45 matlab function sol- ode45 ( (t,y) odefcn (t,y), tspan, y0); Equally splitting time tl linspace (tspan(1),tspan (end),1001) x is the corresponding value for x(1) and x(2) yy1 deval( sol, t1) %plotting y1 (t) vs t for initial condition y (0) 1 dy (0) /dt = 0; figure(1) hold on plot (tl,yy1 (1, :), 'Linewidth',2) title Plot for x(t) vs. t') xlabeltime') ylabelyt) box on 2nd Initial conditions for ode y0 [0;11 Sminimum and maximum time span tspan 0 50 ]; &Solution of ODES using ode45 matlab function sol- ode45 (@ (t,y) odefcn (t ,y), tspan, y0); Equally splitting time t1 linspace(tspan(1), tspan (end),1001) %x is the corresponding value for x(1) and x(2) yy1 deval(sol, t1); %plotting y1 (t) vs t for initial condition &y (0) 0; dy (0)/dt 1; plot (tl,yyl (1, :), 'Linewidth',2) titlePlot for x(t) vs. t') xlabel'time') ylabelyt) ') box on 3rd Initial conditions for ode y0 [0.5;0.5]; minimum and maximum time span 1

tspan 0 &Solution of ODES using ode45 matlab function sol- ode45 (@ (t,y) odefcn (t,y), tspan, y0); Equally splitting time t1 linspace(tspan(1), tspan (end),1001) %x is the corresponding value for x(1) and x (2) yy1 deval(sol, t1); 50 ] 8plotting yl (t) vs t for initial condition y (0) 0.5; dy(0)/dt -0.5; plot (tl,yyl (1, :), 'Linewidth',2) titlePlot for y (t) vs. t for question 3.') xlabel'time') ylabel yt) ' box on $4th Initial conditions for ode y0=[-1;-1 %minimum and maximum time span tspan-[0 50 ]; %Solution of ODES using ode45 matlab function sol- ode45 ( (t,y) odefcn (t,y), tspan, y0) Equally splitting time tl linspace(tspan(1),tspan(end),1001) %x is the corresponding value for x(1) and x (2) yyl deval(sol, t1); 8plotting yl (t) vs t for initial condition y(0) 0.5; dy(0)/dt 0.5; plot (t1,yy1(1, :), 'Linewidth',2) title'Plot for y(t) vs. t for question 3.) xlabel'time' ylabel'yt) ' box on legend'y(0) =1; dy (0) /dt=0', 'y(0)=0; dy(0)/dt 1',... 'y (0) 0.5; dy ( 0) /dt 0.5,'y(0)=-1; dy ( 0 )/dt=-1') fprintfFor Question 3. \n') fprintfAt t tends to infinity it will be oscillating with as sin(t).n') Answering question 4 1st Initial conditions for ode y0 11 a 9.5 9.25 9.125 91; for ii-1:1ength (a) 8minimum and maximum time span tspan 0 150]; Solution of ODES us ing ode45 matlab function sol- ode45 (@ (t,y) odefunc(t, y, a(ii)), tspan, y0); SEqually spl itting time 2

tl linspace(tspan1),tspan (end),1001) x is the corresponding value for x(1) and x(2) yy1 deval (sol, tl); for initial condition %plotting y1 (t) vs %y(0)-1; dy (0)/dt 0; figure (ii+1) ), 'Linewidth',2) plot (tl, Yy not for y(t) vs. t for a-2.2f',a(ii) ) ) titl xlabeltime ylabel'y(t) ') box on end fprintf'\t For question 4. a 9 shows resonance. \n') 8--. Function for evaluating the ODE for question 3. function dydt = odefcn (t,y) eqi y(2); eq2 -y1-10*y(2) +sin(t) Evaluate the ODE for our present problem dydt eq1; eq2]; end Function for evaluating the ODE for question 4. function dydt = odefunc(t,y,a) y(2) a*y (1) +sin (3*t) eqi eq2 Evaluate the ODE for our present problem dydt eql;eq2] end For Question 3 At t tends to infinity it will be oscillating with as sin(t) For question 4. a-9 shows resonance

Plot for y(t) vs. t for question 3. yO) -1 : dv(OVdt 0 v0-0: d( OVdt=1 0.5 0.5 50 10 45 1.5 35 30 25 20 15 10 time Plot for yt vs. t for a 9.50 150 100 50 time 4

Plot for yt vs. t for a-9.25 150 100 50 time vs. t for a 9.12 Plot for y 15 10 5 10 15 150 100 20 50 time 5

Plot for y(t vs. t for a-9.00 25 20 15 5 -5 10 15 20 150 100 50 time Published with MATLAB R2018a 6

%%Matlab code for solving ode
clear all
close all
%Answering question 3
%1st Initial conditions for ode
y0=[1;0];

        %minimum and maximum time span
        tspan=[0 50];
        %Solution of ODEs using ode45 matlab function
        sol= ode45(@(t,y) odefcn(t,y), tspan, y0);
        %Equally splitting time
        t1 = linspace(tspan(1),tspan(end),1001);
        %x is the corresponding value for x(1) and x(2)
        yy1 = deval(sol,t1);
      
        %plotting y1(t) vs t for initial condition
        %y(0)=1; dy(0)/dt=0;
        figure(1)
        hold on
        plot(t1,yy1(1,:),'Linewidth',2)
        title('Plot for x(t) vs. t')
        xlabel('time')
        ylabel('y(t)')
        box on
      
      
%2nd Initial conditions for ode
y0=[0;1];

        %minimum and maximum time span
        tspan=[0 50];
        %Solution of ODEs using ode45 matlab function
        sol= ode45(@(t,y) odefcn(t,y), tspan, y0);
        %Equally splitting time
        t1 = linspace(tspan(1),tspan(end),1001);
        %x is the corresponding value for x(1) and x(2)
        yy1 = deval(sol,t1);
      
        %plotting y1(t) vs t for initial condition
        %y(0)=0; dy(0)/dt=1;
     
        plot(t1,yy1(1,:),'Linewidth',2)
        title('Plot for x(t) vs. t')
        xlabel('time')
        ylabel('y(t)')
        box on
      
%3rd Initial conditions for ode
y0=[0.5;0.5];

        %minimum and maximum time span
        tspan=[0 50];
        %Solution of ODEs using ode45 matlab function
        sol= ode45(@(t,y) odefcn(t,y), tspan, y0);
        %Equally splitting time
        t1 = linspace(tspan(1),tspan(end),1001);
        %x is the corresponding value for x(1) and x(2)
        yy1 = deval(sol,t1);
      
        %plotting y1(t) vs t for initial condition
        %y(0)=0.5; dy(0)/dt=0.5;
     
        plot(t1,yy1(1,:),'Linewidth',2)
        title('Plot for y(t) vs. t for question 3.')
        xlabel('time')
        ylabel('y(t)')
        box on
%4th Initial conditions for ode
y0=[-1;-1];

        %minimum and maximum time span
        tspan=[0 50];
        %Solution of ODEs using ode45 matlab function
        sol= ode45(@(t,y) odefcn(t,y), tspan, y0);
        %Equally splitting time
        t1 = linspace(tspan(1),tspan(end),1001);
        %x is the corresponding value for x(1) and x(2)
        yy1 = deval(sol,t1);
      
        %plotting y1(t) vs t for initial condition
        %y(0)=0.5; dy(0)/dt=0.5;
     
        plot(t1,yy1(1,:),'Linewidth',2)
        title('Plot for y(t) vs. t for question 3.')
        xlabel('time')
        ylabel('y(t)')
        box on      
        legend('y(0)=1; dy(0)/dt=0','y(0)=0; dy(0)/dt=1',...
            'y(0)=0.5; dy(0)/dt=0.5','y(0)=-1; dy(0)/dt=-1')
        fprintf('For Question 3.\n')
        fprintf('At t tends to infinity it will be oscillating with as sin(t).\n')

      
%Answering question 4.
%1st Initial conditions for ode
y0=[1;1];
a=[9.5 9.25 9.125 9];

for ii=1:length(a)
        %minimum and maximum time span
        tspan=[0 150];
        %Solution of ODEs using ode45 matlab function
        sol= ode45(@(t,y) odefunc(t,y,a(ii)), tspan, y0);
        %Equally splitting time
        t1 = linspace(tspan(1),tspan(end),1001);
        %x is the corresponding value for x(1) and x(2)
        yy1 = deval(sol,t1);
      
        %plotting y1(t) vs t for initial condition
        %y(0)=1; dy(0)/dt=0;
        figure(ii+1)
     
        plot(t1,yy1(1,:),'Linewidth',2)
        title(sprintf('Plot for y(t) vs. t for a=%2.2f',a(ii)))
        xlabel('time')
        ylabel('y(t)')
        box on
end

fprintf('\tFor question 4. a=9 shows resonance.\n')
%-----------------------------------------------------------------%

%Function for evaluating the ODE for question 3.
function dydt = odefcn(t,y)

    eq1 = y(2);
    eq2 = -y(1)-10*y(2)+sin(t);
  
    %Evaluate the ODE for our present problem
    dydt = [eq1;eq2];
end
%-----------------------------------------------------------------%

%-----------------------------------------------------------------%

%Function for evaluating the ODE for question 4.
function dydt = odefunc(t,y,a)

    eq1 = y(2);
    eq2 = -a*y(1)+sin(3*t);
  
    %Evaluate the ODE for our present problem
    dydt = [eq1;eq2];
end
%-----------------------------------------------------------------%