3: Math 215 February 2016 Due: Friday Feb 12 in class 1. Analytically, nd the solution y(t) to y = 3 1 2 2 y, with y(0) = 1 1 . Use Matlab to plot the direction eld for this system. Overlay your analytic solution for this particular initial condition on the direction eld. You may want to consult the other homework le which shows how to use Matlab for direction elds and plotting analytic and numerical solutions. 2. Solve the following system of dierential equations with initial conditions and sketch the solution. Check that you can do this by hand as best you can (of course you can use Matlab to verify that the key features of your sketch are correct). On your sketch, label the initial point, initial velocity vector and asymptotic direction as t . 2 2 0 x = x x(0) = 4 7 1 3. We consider a system of dierential equations containing the parameter a. 0 1 x = 5 a x (a) Determine the eigenvalues in terms of a. (b) Find the critical value(s) of a where the qualitative nature of the eigenvalues changes and summarize your ndings in words. (c) Use Matlab to produce a direction eld plots for each possible qualitative behaviour of this system and in each case plot several solutions on the direction eld. Label the plots with the value of a that you used and the description of the xed point at (0, 0) (e.g. \"Stable node\