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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' ;108642 2 4 6 8 10108642 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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This note was uploaded on 06/28/2009 for the course MATH 246H taught by Professor Zheng during the Spring '08 term at Maryland.
 Spring '08
 ZHENG
 Differential Equations, Equations

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