Question

MATLAB HELP

(a) Use the command dsolve to find general solutions to the differential equations below. i. y 00 + 3y = 0 ii. y 00 + 4y 0 + 29y = 0 iii. y 00 − y/36 = 0 iv. y 00 + 2y 0 + y = 0 v. y 00 + 6y 0 + 5y = 0 (b) Graph each of the solutions in (a) in the same window with 0 ≤ t ≤ 10, using the initial conditions y(0) = 1 and y 0 (0) = 0. (c) Which of these equations does not represent a mechanical vibration? Why not? In your comments, explain how to recognize that the equation cannot describe a mechanical vibration i. from the graph of the solution, and also ii. from the coefficients of the original equation. (Recall how we have interpreted the coefficients m, b, and k where m y00 + b y0 + k y = 0.) (d) Classify the other four solutions as undamped, underdamped, critically damped, or overdamped.

(a) Use the command dsolve to find general solutions to the differential equations below. 1. i. у" + Зу — 0 ii. y"4y29y = 0 ii. у" — у /36 %3D 0 iv. y"2y'y= 0 у"+6у'+ 5у%30 + V. (b) Graph each of the solutions in (a) in the same window with 0 < t < 10, using the initial conditions y(0) 1 and y'(0) = 0. (c) Which of these equations does not represent a mechanical vibration? Why not? In your comments, explain how to recognize that the equation cannot describe a mechanical vibration i. from the graph of the solution, and also ii. from the coefficients of the original equation. (Recall how we have interpreted the coefficients m, b, and where my" by' + ky = 0.) (d) Classify the other four solutions as undamped, underdamped, critically damped, or overdamped.

Answer 1

Matlab code for dsolve for solving ode clear all close all syms y (t) mb k eqn m*diff (y ,t,2) +b*diff (y,t,l) +k*y == 0; iff (Y, t); Dy , Dy(0)--0]; ySol (t,m,b,k) - dsolve (eqn,cond); tt-linspace0,10 ); m-1;b-0; k- 3 ; yy1-ySol (tt, 1,0,3); hold on plot (tt,yy1, linewidth',2) fprintfFor question i. m-1;b 0 ; k - 3 \ n') fprintfhence b 2-4*m*k=% £\n' ,b^2-4*m*k) fprintf('as b*2-4*m*k<0 so it is oscillatory\n') fprintfAnd as b-0 it is undamped \n') fprintfFrom the plot it also shows undamped oscillation.\n\n' ) yy2=ySol (tt,1,4,29) ; hold on plot (tt,yy 2, linewidth',2) m 1;b -4; k-29; fprintfFor question ii. m=1;b - 4; k=29\n' fprintfhence b 2-4 *m*k=% f\n' ,b 2-4*m*k) fprintfas b*2-4*m*k<0 so it is oscillatory\n') fprintfAnd as b is nonzero it is underdamped\n') fprintfFrom the plot it also shows underdamped oscillation.\n\n' yy 3-ySol (tt,1,0,-1/36) hold on plot (tt,yy 3, 'linewidth',2) m 1;b-0;k-1/36; fprintfFor question iii. m 1;b -0 ;k=-1/36\n') fprintfhence b 2-4 *m*k=% f\n ' ,b^2-4*m*k) fprintfas b 2-4*m*k>0 so it is nonoscillatory\n') fprintfAnd as b-zeroo it is nonoscillatory\n' fprintfFrom the plot it also shows non oscillation.\n\n') yy4-ySol (tt, 1,1,2) ; hold on plot (tt,yy4, 'linewidth',2) m 1;b-1;k-2; fprintr fprintfhence b*2-4*m*k=% f\n ' , b^2-4*m*k) fprintfas b 2-4*m*k<0 so it is oscillatory \n') fprintfAnd as b is nonzero it is underdamped\n') fprintfFrom the plot it also shows underdamped oscillation. \n\n' ) m=1;b - 1; k=2\n ' ) iv. 1

yy5-ySol (tt,1,6,5); hold on plot (tt,yy5, 'linewidth',2) m 1;b-6; k- 5; fprintfFor question iv. m=1;b - 6; k=5\n' ) fprintfhence b 2-4*m*k=% £\n' ,b^ 2-4*m*k) fprint s 0 so it is nonoscillatory\n') грrint fprintfFrom the plot it also shows overdamped oscillation. \n\n') xlabel'time t') ylabelyt) ') title'y(t) vs. time plot') legendques (i), 'ques (ii), 'ques (iii),'ques (iv), 'ques (v)') 88888888 8888 End of Code %%8$%%88%%9%8% For question i. m=1;b-0; k =3 hence b 2-4 m*k=-12.000000 as b 2-4*m*k<0 so it is oscillatory And as b-0 it is undamped From the plot it also shows undamped oscillation. For question ii. m=1;b=4;k-29 hence b 2-4*m*k=-100.000000 as b 2-4*m*k<0 so it is oscillatory And as b is nonzero it is underdamped From the plot it also shows underdamped oscillation. For question iii. m=1; b-0; k--1/36 hence b 2-4 *m*k= 0.1111 11 as b 2-4 m*k>0 so it is nonoscillatory And as bzero it is nonoscillatory From the plot it also shows non oscillation For question iv. m=1 ;b-1;k -2 hence b 2-4 *m *k=- 7.000000 as b 2-4 *m*k<o so it is oscillatory And as b is nonzero it is underdamped From the plot it also shows underdamped oscillation. For question iv. m=1 ;b-6;k-5 hence b 2-4 *m*k=16.000000 as b 2-4 *m*k>0 so it is nonoscillatory And as b is nonzero it is overdamped From the plot it also shows overdamped oscillation 2

y(t) vs. time plot ques () ques ( 25 ques (v) 1 5 MA 0.5 0 05 6 9 10 time f Published with MATLAB R2018a

%%Matlab code for dsolve for solving ode
clear all
close all

syms y(t) m b k
eqn = m*diff(y,t,2)+b*diff(y,t,1)+k*y == 0;
Dy = diff(y,t);
cond = [y(0)==1, Dy(0)==0];
ySol(t,m,b,k) = dsolve(eqn,cond);

tt=linspace(0,10);
m=1;b=0;k=3;
yy1=ySol(tt,1,0,3);
hold on
plot(tt,yy1,'linewidth',2)
fprintf('For question i. m=1;b=0;k=3\n')
fprintf('hence b^2-4*m*k=%f\n',b^2-4*m*k)
fprintf('as b^2-4*m*k<0 so it is oscillatory\n')
fprintf('And as b=0 it is undamped\n')
fprintf('From the plot it also shows undamped oscillation.\n\n')

yy2=ySol(tt,1,4,29);
hold on
plot(tt,yy2,'linewidth',2)
m=1;b=4;k=29;
fprintf('For question ii. m=1;b=4;k=29\n')
fprintf('hence b^2-4*m*k=%f\n',b^2-4*m*k)
fprintf('as b^2-4*m*k<0 so it is oscillatory\n')
fprintf('And as b is nonzero it is underdamped\n')
fprintf('From the plot it also shows underdamped oscillation.\n\n')

yy3=ySol(tt,1,0,-1/36);
hold on
plot(tt,yy3,'linewidth',2)
m=1;b=0;k=-1/36;
fprintf('For question iii. m=1;b=0;k=-1/36\n')
fprintf('hence b^2-4*m*k=%f\n',b^2-4*m*k)
fprintf('as b^2-4*m*k>0 so it is nonoscillatory\n')
fprintf('And as b=zero it is nonoscillatory\n')
fprintf('From the plot it also shows non oscillation.\n\n')

yy4=ySol(tt,1,1,2);
hold on
plot(tt,yy4,'linewidth',2)
m=1;b=1;k=2;
fprintf('For question iv. m=1;b=1;k=2\n')
fprintf('hence b^2-4*m*k=%f\n',b^2-4*m*k)
fprintf('as b^2-4*m*k<0 so it is oscillatory\n')
fprintf('And as b is nonzero it is underdamped\n')
fprintf('From the plot it also shows underdamped oscillation.\n\n')

yy5=ySol(tt,1,6,5);
hold on
plot(tt,yy5,'linewidth',2)
m=1;b=6;k=5;
fprintf('For question iv. m=1;b=6;k=5\n')
fprintf('hence b^2-4*m*k=%f\n',b^2-4*m*k)
fprintf('as b^2-4*m*k>0 so it is nonoscillatory\n')
fprintf('And as b is nonzero it is overdamped\n')
fprintf('From the plot it also shows overdamped oscillation.\n\n')

xlabel('time t')
ylabel('y(t)')
title('y(t) vs. time plot')
legend('ques (i)','ques (ii)','ques (iii)','ques (iv)','ques (v)')

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