1. Consider the ODE d 2 '9' y 7 : 2 dx + logrc mlogrc (:r>1) (a) Find the general solution of the ODE. (b) Does the ODE have any equilibrium solutions? Explain your answer. (c) For what initial conditions, if any, is x) a straight line? 2. Carp, Cyprmas comma, are an invasive species of sh which are not native to Australia but now live in many Australian river systems. Carp cause damage to Australian river ecosys- tems, for instance by increasing water turbidity (muddiness) and reducing the abundance of water plants and invertebrates.1 It has been proposed to use a virus, Cyprmid herpesm'ms 3, as a biological control for carp.2 In this question we will model the spread of the virus in a carp population. Let Ht) be the number (in thousands) of carp infected with the virus at time 1' years after the virus is released, and let N be the total number of sh (in thousands) in the population. The rate of spread of the virus is proportional to the number of sh that are currently infected, and the number of sh that are not yet infected. Once a sh becomes infected, it remains infected and does not recover. For the purposes of this question, we will assume the virus does not kill the sh. Therefore Mi AgNi where 5 and N are positive constants. (a) Find the equilibria of the ODE, draw a phase plot, and sketch the family of solutions. Interpret your results in terms of the virus's spread through the carp population. (b) Let 6 : and N : 10,000. Suppose that the virus outbreak begins with 1000 infected sh. Find how long it will take until 90% of the sh are infected