4. Consider the IVP t y +1 y(0) = yo- (a) Without solving, determine all values of yo for which the IVP is certain to have a unique solution. (b) Solve the IVP for all such values of yo and detemine the interval of validity for them. (c) What can be said about the solution of the IVP for other values of yo? Part II - Autonomous ODES 1. Given the differential equation y = y - 3y + 2 (a) Find all equilibrium points, and sketch the phase line. (b) Classify the type of the equilibrium solutions (stable, semistable or unstable). (c) If y = y(t) represents the position of a particle, what are the velocity and acceleration of the particle in terms of y? In particular, what are the velocity and acceleration of the particle when it is found at y = 0? (d) Where does the particle have its largest/smallest value of its velocity/acceleration? (e) Sketch the equilibrium solutions along with a few integral curves above, below or between the equilibrium solutions. (f) Concavity of the integral curves are measured by acceleration as you found above. Are there any inflection levels, crossing through which the concavity of the integral curves changes? 1 (g) Let y(t) be the solution of the ODE satisfying the initial condition y(0) = 4/3. Find the limits of y(t) as to and as t- (for this you do not need to find y(t) explicitly). (h) Find all yo such that the solution of the ODE with the initial condition y(0) = yo has the same limit at o as the solution from the previous item. (i) Let y(t) be the solution of the ODE with y(0) = 3. Decide whether y(t) is monotonically decreasing or increasing and find to what value it approaches when t increases (the value might be infinite). (j) Find the solution of the ODE with y(0) = 3 explicitly. Determine the interval of validity of the IVP.