(10 points) A NASA spacecraft moves in an x, y plane subject to the 2 x 2 LODE System x' = Ax where A = , oo < t < 00. (I) x(t) = Entries of A indicate thrusts of the rocket engines. of A, with VI = 1. (a) Determine the E-val and E-vec v = , X2(t) = (b) Determine a fundamental pair of solutions Xl(t) = Hint: Check that the generalized E-vec w = v = (A A1)w can be chosen so that WI = 0, u,'2 = 1/5. (c) State the Superposition Principle for system (1). (d) Determine the general solution written as a pair of scalar equalities for entries c(t) and y(t). Name the type of the critical point O indicating its stability/instability. (e) Draw a phase portrait including arrows directing the motion. (One so- lution curve from each of the half-planes would suffice, plus the eigen-vector line.) and with x(to) = (f) Solve the IVP for (I) with x(to) = = 4.8. Calculate the velocity vectors x' (to) for the IVP solutions. where to (g) Someone from Penn State Engineering guesses that in fact, c(t) satisfies a 2nd order LODE + ac' + bc = O. Check if the guess is correct and specify coeffcients a and b. (h) Solve the ODE specified in (g) and reproduce the solution x(t) from (d). (i) For the general solution from (d) calculate the limits of as t i oo and as t 00. Confirm or dispute that the solution curves are asymptotically parallel or perpendicular to the E-vec direction. (j) In the reversed time, corresponds to a spring-mass system that is damped.