July 19, 2002 Math 221 Test 1 Summer(2) 2002Name: PIN(in any 4 digits): Score:Instructions: You must show supporting work to receive full and partial credits. No text book, notes, formulasheets allowed.1(10pts) (a) Is (3t − t3+ 1)1/3the solution to this IVP:dxdt=1 − t2x, x(0) = 1? Verify your answer.(b) Determine whether or not x2y3− 2x3y2= C defines an implicity solution to the equation (2y2− 6xy)dydx+(3xy − 4x2) = 0.2(10pts) Find a general solution to the equation ydydx= x3e−y2.3(20pts) Consider the linear equationdxdt=xt + 1+ 2t2− 2.(a) Find a general solution to the homogeneous equation.(b) Find a particular solution to the nonhomogeneous equation.(c) Find the solution that also satisfies the initial condition x(0) = 1.4(15pts) Does the Existence and Uniqueness Theorem apply to this initial value problemdydx= xy1/2, y(1) = 0? If not,why not? And if not, find at least 2 solutions.5(15pts) Consider the IVP: x0=x2+ tx, x(0) = −1.(a) Use a step size h = 0.25 and the Euler method to approximate the solution in the interval [0, 1].(b) Sketch your approximating solution.6(15pts) (a) The vector field of an equation is given below. Sketch solutions that go through these points: (i) (−2, 0),(ii) (0, 5).−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−10123456(b) Consider the equation x0= −x + t. Use the isocline method to sketch a solution portrait of the equation(i.e. a few isoclines of your choice, vector field on the isoclines, and a few typical solution curves). Based onyour analysis, what can you conclude about the limit limt→∞x(t) for any solution x(t)? If you concludedthat limt→∞x(t) = ∞, would x(t) tend to infinity faster than t2as t → ∞? Justify your answer.7(15pts) Consider the autonomous equationdxdt= f(x) = x4− 3x3+ 2x2.(a) Sketch the phase line and classify each equilibrium point as sink, source, or node,(b) Let x(t) be a solution satisfying x(1) = 0.5. Then what are the limits of limt→∞x(t) and limt→−∞x(t)?ENDMath 221 Test 1, Summer(2) 2002 Page