Differential Equations

Draw on paper the solution x(t) = v1 e?1t + v2 e?2t on the x1x2-plane and then answer the questions below.(a) Based on your graph, select the correct solution curve from the interactive graph below.• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None(b) Use the graph below to find lim t?+??x(t)?, where ?x(t)? is the length of the solution vector x(t).• Select One • Zero • Infinity • None(c) Introduce the unit vectors u1 = v1/?v1? and u2 = v2/?v2?. Use the graph below to find thelim t?+?u(t), where u(t) = x(t) ?x(t)?, is a unit vector in the direction of the solution vector x(t).[Select One/U1/-U1/U2/-U2/None](d) Use the graph below to find lim t????x(t)?.• Select One • Zero • Infinity • None(e) Use the graph below to find the lim t???u(t), where u(t) = x(t) ?x(t)?[Select One/U1/-U1/U2/-U2/None]9(f) Characterize the zero solution, x0 = 0.• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • NoneComments on the graph below: • The graph is interactive. • You can click on the boxes to turn on and off possible solution curves. • For each possible solution we display: the possible solution curve, the possible solution vectorx(t), and the associated unit vector u(t) = x(t)/?x(t)?. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)change in time. • You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see howthe curves would change.

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