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 Practice Midterm #3 Notation. � = d/dt. 1. (a) If f ∈ E and F (s) = L[f(t)], show that lims→∞ F (s) = 0. s + 1 (b) Find the inverse Laplace transform of F (s) = log . s − 1 2. (a) Sketch the graph of f(t) = (1/5)(h(t − 5)(t − 5) − h(t − 10)(t − 10)), where h(t) is the unit step function or the Heaviside function. (b) Find the solution of the initial value problem y�� + 4y = f (t), y(0) = 0, y�(0) = 0. Sketch the graph of the solution. (c) Compute the left and the right limits of y��(t) at t = 5 and t = 10. 3. Consider two vectors �y1(t) = (t, 1) and �y2(t) = (t2 , 2t). (a) In which intervals are �y1 and �y2 are linearly independent? (b) Find a system of differential equations satisfied by �y1 and �y2. 4. Find the general solution of � � � �� � x � 1 −1 x y =1 3 y. 5. Let � � 0 1 A = . −1 0 (a) Show that A2 = −I. (b) Show that � � At cos t sin t e = . − sin t cos t (c) Find the general solution of x � x = A . y y (d) Sketch solutions in the (x, y)-plane and discuss their behavior.
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