Solutions10 - Math 601 Solutions to Homework 10 1. Consider...

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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:ddtxy=222-1‚ •xyTo 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:nullspaceλ1-2-2-2λ1+ 1=nullspace1-2-24=nullspace1-200=Span21‚¶Thus, the vector21is an eigenvector associated with the eigen-valueλ1= 3.
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Chapter 9 / Exercise 11-14
Precalculus: Mathematics for Calculus
Redlin/Stewart
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Forλ2=-2, we have:nullspaceλ2-2-2-2λ2+ 1=nullspace-4-2-2-1=nullspace2100=Span1-2‚¶Thus, the vector1-2is an eigenvector associated with theeigenvalueλ2=-2.Thus, the general solution to the system of linear differential equa-tions is:xy=c1e3t21+c2e-2t1-2We 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

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Term
Spring
Professor
alndy
Tags
Equations, Parametric Equations, Eigenvalue eigenvector and eigenspace, dr d, Parametric equation
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Precalculus: Mathematics for Calculus
The document you are viewing contains questions related to this textbook.
Chapter 9 / Exercise 11–14
Precalculus: Mathematics for Calculus
Redlin/Stewart
Expert Verified

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