Assignment 6

# Assignment 6 - x(0 = 4,y(0 = 4(d Draw the x and...

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Professor Jason Levy, University of Ottawa, MAT 1332C, Winter 2011 Assignment 6, due Wednesday April 6, 8:30am at the beginning of class. Late assignments will not be accepted; nor will unstapled assignments. Student Number DGD section By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. Signature Student Name 1. Find the tangent plane to the graph of the function f ( x,y ) = 2 π - 3 cos 4 x + 3 sin 2 y at the point (0 ). 2. Find the (2,2)-entry of the Jacobian matrix of the function F ( x,y ) = ± x 2 e y + 2 x sin( x y ) sin( x 2 ) - 3 ye - x ² at the point (2 , 1) . 3. Consider the following system of linear diﬀerential equations: dx dt = - 3 x + y dy dt = 4 x - 3 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) = 4 ,y (0) = 4 . (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. 4. Consider a disease that propogates according to the system dx dt = 16-. 2 xy-. 4 x dy dt = 0 . 1 xy-8 y 1 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 ±-. 4-. 2 y-. 2 x . 1 y . 1 x-8 ² (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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Assignment 6 - x(0 = 4,y(0 = 4(d Draw the x and...

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