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PROBLEMS 1. Solve the equation dy dr 4+ga Also. find the interval of validity (i.e. of existence) for the solution passing through the point (0,1). 2. Solve dy 043 dt where ando are positive constants. Note that this equation shows up in the study of the stability of fluid flow. 3. State the definition of a norm L on a real vector space V. Let V = C[O, 1] and L be defined by L(f) max |/(x)|. =E[0,1] Using L. find the distance between ! and g. where f(x) x4 and 4. Consider the sequence To = 1, In+1=1+; 1+m Show that .r', is Cauchy in @ {set of rational numbers} but does not converge Q 5. Given the system dy f(t,y) for (t,y)E(a,b)x D (1) y(to = yo- (a) What two conditions on are needed to guarantee that (1) will have unique solution? (b) Consider the following model for viral infection V'=aY-bV, (2) X'=c-dX-BXV where V is the population of virions, x is the population of healthy cells,and) representsthe population of infected cells and a.b. are positive constants. Argue that for the initial state (Vo. Xg.0). where Vo Xa > 0. model (2) has s unique positive solution. 6. According to Kirchhoff's Second Law. specific RI. circuit with no driving electromotive force can be modeled as follows: (3) +2+242 where Q(E) is the charge on the capacitor at time = 0, (a) Write equation (3) as a : dimensional first order system in z and y. Start with =2. (b) Find the eigenvalues of the system. (c) Draw the solution curve on the phase plane with initial condition (3) 7. Suppose ¿f is an eigenvector of a 2x2 matrix A corresponding to eigenvalue A. solution of (A-Af)eid if. Show that and & are linearly independent 8. Consider the following linear system: (4) (a) Find the general solution T(E) of system (4). (b) Argue that I= is the unique equilibrium of the system. (c) Calculate lime +00 I(()) and use this to decide whether Ois unsta- ble stable. (Disclaimer: The above list of material and problems is not exhaustive as you should have in your notes ideas and problems have stressed in class and you should have working knowledge of the homework you have turned in.)

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