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 Final Notation. � = d/dt. 1. (a) Is the differential form x2y3dx + x(1 + y2)dy exact? (b) Find a function µ(x, y) so that µ(x, y)(x2y3dx + x(1 + y2)dy) becomes exact. (c) Solve the differential equation dy x2y3 . dx = −x(1 + y2)2. (a) A basis of solutions of the differential equation 2t 1 y�� + t2 − 1y� − 16(t2 − 1)2 y = 0 is given by � �2 � �2 y1(t) = t − 1 , y2(t) = t + 1 . t + 1 t − 1 Compute the Wronskian W (y1, y2). (b) Use variation of parameters to find a particular solution of 2t 1 2 y�� + t2 − 1y� − 16(t2 − 1)2 y = t − 1. 3. Show that the initial value problem y� = |y|1/2 and y(0) = −1 is well-posed on t ∈ [0, a) if a ≤ 2 but not if a > 2. 4. Solve the initial value problem � � � �� � � � � � x � 1 4 x x(0) 2 y = 1 1 y, y(0) = 1 (a) by using the eigenvalues and (b) by the Laplace transform. 5. Consider the plane autonomous system x� = y, y� = x 2 − y − �, where � is a real parameter. (a) Find the critical points. (b) If the system has critical points, then discuss the behavior of the solutions near the critical points. (c) Discuss how the behavior of solutions changes with �. 6. (a) The Li´enard euqation is u�� + c(u)u� + g(u) = 0, where c(u) ≥ 0 and g(0) = 0, ug(u) > 0 for u = 0 � and small. Show that the critical point u = 0 and u� = 0 is stable. (b) When c(u) = 1, show that the critical point u = 0 and u� = 0 is asymptotically stable. 17. (a) Show that every solution of the plane autonomous system x� = ye 1+x2+y2 , y� = −xe 1+x2+y2 is periodic. (b) Show that the system x� = x − x 3 − xy 2 , y� = y − y 3 − yx 2 has a unique limit cycle. (c) Show that the system x� = x − xy 2 + y 3 , y� = 3y − yx 2 + x 3 has no nontrivial periodic solution lying inside the circle x2 + y2 = 4.
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