Math 601 Solutions to Homework 101. Consider the vector fieldF(x, y) = (2x+ 2y)i+ (2x-y)j.(a) Find the parametric equations for the flow line ofFbeginning atthe point (0,5).(b) Find div(F).(c) Find rot(F).Answer:(a) We need to solve the system of differential equations:dxdt=2x+ 2ydydt=2x-yThis is a system of linear differential equations, which we can writeas:ddt•xy‚=•222-1‚ •xy‚To solve this system, we need to compute the eigenvalues andeigenvectors of the matrix.We can quickly find the eigenvalues if we remember that the prod-uct of the eigenvalues must equal the determinant of the matrix,and the sum of the eigenvalues must equal the trace of the matrix.So,λ1λ2=-6 andλ1+λ2= 1. By inspection we see that theeigenvalues areλ1= 3 andλ2=-2. (This method only works for2×2 matrices.)Now, we find the eigenvectors associated with each eigenvalue.Forλ1= 3, we have:nullspace•λ1-2-2-2λ1+ 1‚=nullspace•1-2-24‚=nullspace•1-200‚=Span•21‚¶Thus, the vector•21‚is an eigenvector associated with the eigen-valueλ1= 3.
Forλ2=-2, we have:nullspace•λ2-2-2-2λ2+ 1‚=nullspace•-4-2-2-1‚=nullspace•2100‚=Span•1-2‚¶Thus, the vector•1-2‚is an eigenvector associated with theeigenvalueλ2=-2.Thus, the general solution to the system of linear differential equa-tions is:•xy‚=c1e3t•21‚+c2e-2t•1-2‚We would like to find the solution which begins at the point (0,5),so we wantx= 0, y= 5 whent= 0. We can plug those num-bers into the above equation and solve forc1andc