Question

Question 5: Find the exact solutions in the time domain using dsolve. Recall the matrix representation for our two-vessel water clock is: d又 -=Ax, where A= DE: Hint: Start with the code below. Then enter A, x0, and the DE. Use dsolve. syms yl (t) y2 (t) Use dot notation to access the components of the solution Question 5: Record your solution for the two unknowns y1(t) and y2(t). The first is given for you y2 (t) = y, (t) = e-t, Answer: Question 6: Cumulative Outflows Let's now focus on the col Let fi(t) and f2(t) be the cumulative outflows from time 0 to time t, for each vessel. Because of the identity: and the total outflows from each vessel Qi = 1 , y(t) these integrated outflows are just: ) ) dt and fa(t) 2() dt Evaluate each integral using MATLAB's int command, then create a simultaneous plot of the three quantities: Hideal (t) = t, fi (t) and f2 (t) = H(t). The plot is given for free. Question 6: Record your answers for the cumulative outflows. The first is given for free. f1 (t) = 1-e-t, f, (t) = Answer:

Answer 1

%%Matlab code for solving set of eqa tions using dsolve clear all close all Answering question 5. Solving system of equation using dsolve syms yl(t) y2(t) eqns[diff (yl, t)---yl, diff (y2,t)yl-y2]; cond-y1 (0), y2(0)1: [ y 1Sol ( t ) , y2 Sol ( t ) ] -dsolve ( eqns , cond ) ; %printing the result fprintf(Solution using dsolve\n); fprintf(yl(t)) disp(ylSol) fprintf( y2(t)- disp (y2Sol) Answering question 6. %calculating cumulative outflow yl_int-int (ylSol,t, [0 t]); y2_int-int (y2Sol,t,[0 t]) printing the result fprintf 'Solution using intln'; fprintf( 'fl(t) disp (yl_int) fprintf E2(t) disp (y2_int) 용동号%%号号%%号号%%号号%%号号 End of Code 욤응응용용융응용용융융용용융융용용융융용 Solution using dsolve yl (t)xp(-t) symbolic function inputs: t y2(t) exp(-t)+t*exp(-t) symbolic function inputs: t Solution using int fl (t)-l- exp(-t) f2 (t)-2 exp (-t)*(t 1) -exp(-t) + - Published with MATLAB R2018a

%%Matlab code for solving set of eqations using dsolve
clear all
close all
%Answering question 5.
%Solving system of equation using dsolve

syms y1(t) y2(t)
eqns = [diff(y1,t)==-y1, diff(y2,t)==y1-y2];
cond = [y1(0)==1, y2(0)==1];

[y1Sol(t),y2Sol(t)] = dsolve(eqns,cond);
%printing the result
fprintf('Solution using dsolve\n');

fprintf('y1(t)=')
disp(y1Sol)

fprintf('y2(t)=')
disp(y2Sol)

%%Answering question 6.
%calculating cumulative outflow
y1_int=int(y1Sol,t,[0 t]);
y2_int=int(y2Sol,t,[0 t]);

%printing the result
fprintf('Solution using int\n');

fprintf('f1(t)=')
disp(y1_int)

fprintf('f2(t)=')
disp(y2_int)

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