Final answer:
To sketch the phase portrait for the given system x'=Ax+(-1,3), we first find the matrix A based on the given information about its eigenvalues and eigenvectors. Then, we use the matrix A to determine the general solution for the system. Finally, we can plot x1(t) for the initial condition x(0)=(3,2).
Step-by-step explanation:
In order to sketch the phase portrait for the system x′=Ax+(−1,3), we need to find the matrix A first. We are given that A(2,1) = (1,-3), which means that the first column of A is (1,-3). We are also given that A has an eigenvalue λ=-3+2i and corresponding eigenvector v=(1+3i,i). Since the matrix A is not given, we can use the fact that the matrix A is similar to its Jordan form, which is a matrix that has 1s on the superdiagonal and has the eigenvalues on the diagonal. Therefore, A is similar to the matrix J = [(−3+2i, 0), (0, −3−2i)].
Now that we have the matrix A, we can find the general solution for x′=Ax+(−1,3). Let λ1 = -3+2i and λ2 = -3-2i be the eigenvalues of A. Let v1 and v2 be the eigenvectors corresponding to λ1 and λ2, respectively. The general solution is given by x(t) = c1*e^(λ1*t)*v1 + c2*e^(λ2*t)*v2, where c1 and c2 are constants determined by the initial condition.
Finally, to sketch x1(t) for the initial condition x(0)=(3,2), we can plug in the values of t and the eigenvectors into the general solution x(t) = c1*e^(λ1*t)*v1 + c2*e^(λ2*t)*v2 and plot the resulting values of x1(t).