MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.034 Midterm #1 Name: 1. (20 points) Solve the initial value problem y�y�� − t = 0, y(1) = 2, y�(1) = 1. 118.034 Midterm #1 Name: 2. Consider the differential equation y� = y(5 − y)(y − 4)2 . (a) (7 points) Determine the critical points (stationary solutions). (b) (5 points) Sketch the graph of f(y) = y(5 − y)(y − 4)2 . (c) (8 points) Discuss the stability of critical points in part (b). 218.034 Midterm #1 Name: 3. (20 points) Determine the values of a, if any, for which all solutions of the differential equation y�� − (3 − a)y� − 2(a − 1)y = 0 tend to zero as t → ∞. Here, � = d .dt 318.034 Midterm #1 Name: 4. Consider the undamped forced vibration system y�� + y = 3 cos ωt, y(0) = 0, y�(0) = 0. (a) (10 points) Find the solution for ω = 1� . (b) (5 points) Find the solution for ω = 1. (c) (5 points) Discuss the behavior of solutions in part (a) and part (b). 418.034 Midterm #1 Name: 5. (a) (5 points) State the Sturm Comparison Theorem. (b) (15 points) Show that no nontrivial solution of y�� + (1 − t2)y = 0 vanishes infinitely often on 0 < t < ∞.
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