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MAT1332W10A6 - x(0 = 2,y(0 =-2(d Draw the x and...

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Calculus for the life Sciences II MAT 1332B, MAT 1332C Assignment 6 ( Optional ) Due date: 10:00 in class, April 12, 2010 Instructor (circle one): Robert Smith? , Jing Li DGD (circle one): 1, 2, 3, 4 Student Name (printed) Student Number By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. If you include additional pages, please ensure they are stapled and that your name is on every page. Signature 1. Find the tangent plane to the graph of the function f ( x, y ) = 3 π - 3 cos 2 x + 2 sin 3 y at the point (0 , π ). 2. Find the (2,2)-entry of the Jacobian matrix of the function F ( x, y ) = x 2 y + 2 xe y x/y - 3 ye - x at the point (2 , 1) . 3. Consider the following system of linear differential equations: dx dt = - x + y dy dt = 4 x - y (a) Find the eigenvalues and eigenvectors associated with the system. (b) Write down the general solution formula for the system. (c) Give the particular solution for the initial values
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Unformatted text preview: x (0) = 2 ,y (0) =-2 . (d) Draw the x- and y-nullclines and the direction arrows in the phase plane. (e) Sketch the solution curve for the initial condition in part (c) into the phase plane. (f) Is the point (0,0) stable or unstable? Classify this equilibrium. 1 4. Consider a disease that propogates according to the system dx dt = 12-. 1 xy-. 3 x dy dt = 0 . 1 xy-6 y where x represents susceptible individuals, y represents infected individuals. (a) Find all biologically meaningful steady states. (b) Show that the Jacobian matrix of this system is given by •-. 3-. 1 y-. 1 x . 1 y . 1 x-6 ‚ (c) For the biologically meaningful steady states from (a), find the eigenvalues of the Jacobian matrix. (d) Determine the stability of the biologically meaningful steady states. 2...
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MAT1332W10A6 - x(0 = 2,y(0 =-2(d Draw the x and...

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