Problem Set F Numbers 1 through five

Problem Set F Numbers 1 through five - Problem Set F...

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Unformatted text preview: Problem Set F Numbers 1,5,7 1. a) A = [2 -1;3 -2;]; [xi,R] = eig(sym(A)) Eigenvalue: 1 Eigenvector : [1, 1] Eigenvalue: -1 Eigenvector: [1/3, 1] General Solution: = + /- y c111et c21 31e t A2 = [1 -1; 5 -3]; [xi,R] = eig(sym(A2)) Eigenvalue: [-i -1] Eigenvector: [2/5 - i/5, 1] Eigenvalue:[i 1] Eigenvector[i/5 + 2/5, 1] General Solution: u(t) = - ( + ( ) e t cost sin t ) A3 = [-3 5/2;-5/2 2]; [x3i R] = eig(sym(A3)) %finding a second eigenvector for A3 M = [-5/2 5/2; -5/2 5/2]; eta = M\x3i Eigenvalue:[-1/2] Eigenvector: [1,1] Eigenvalue: [-1/2] Eigenvector[-2/5, 0] General Solution: = - / + [- +- /- / ] y c111e 1 2t c2 11te 12t 2 50e 1 2t b) %Finding the general solution using dsolve ivp = 'Dx = 2*x -y, Dy =3*x - 2*y, x(0) = a, y(0) = b' ; [x y] = dsolve(ivp , 't' ) x = exp(t)*((3*a)/2 - b/2) - ((3*a)/2 - (3*b)/2)/(3*exp(t)) y = exp(t)*((3*a)/2 - b/2) - ((3*a)/2 - (3*b)/2)/exp(t) %graphing the solution curves given by dsolve xf = @(t,a,b)eval(vectorize(x)); %defining the functions for xf and yf yf = @(t,a,b)eval(vectorize(y)); figure; hold on t = -3:0.1:3; for a = -2:2 %the for loops, graph several trajectories using different sets of initial condititions for b = -2:2 plot(xf(t,a,b),yf(t,a,b)) end end hold off axis([-10 10 -10 10]) xlabel 'x' ylabel 'y' title 'graph of the ivp Dx = 2*x -y, Dy =3*x - 2*y, x(0) = a, y(0) = b' ;-10-8-6-4-2 2 4 6 8 10-10-8-6-4-2 2 4 6 8 10 x y graph of the ivp Dx = 2*x -y, Dy =3*x - 2*y, x(0) = a, y(0) = b From the graph the origin appears to be an unstable saddle point also the vector field shows the direction that t is increasing. ivp = 'Dx = 1*x -y, Dy =5*x -3*y, x(0) = a, y(0) = b' ; [x y] = dsolve(ivp , 't' ) x = - exp(t*(i - 1))*((5*a*i)/2 - b*(i + 1/2))*(i/5 + 2/5) - (((5*a*i)/2 - b*(i - 1/2))*(i/5 - 2/5))/exp(t*(i + 1)) y = ((5*a*i)/2 - b*(i - 1/2))/exp(t*(i + 1)) - exp(t*(i - 1))*((5*a*i)/2 - b*(i + 1/2)) %Graphing trajectories of the solution for varying initial conditions xf = @(t,a,b)eval(vectorize(x)); yf = @(t,a,b)eval(vectorize(y));...
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Problem Set F Numbers 1 through five - Problem Set F...

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