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 #2 Name: 1. (a) (10 points) Find a linear differential equation with constant coefficients that has solutions t, et and tet . (b) (10 points) Prove or disprove that t4 and t6 can be solutions of one and the same linear ho-mogeneous differential equation y�� + p(t)y� + q(t)y = 0 on an interval [−1, 1], where p and q are continuous on [−1, 1]. 118.034 Midterm #2 Name: 2. For a certain regular linear differential operator L a basis of solutions of Ly = 0 is given by y1(t) = e 2t , y2(t) = 2t2 + 2t + 1. (a) (10 points) Compute the Wronskian of y1 and y2. (b) (10 points) Using variation of parameters, find a solution of the differential equation Ly = t2e2t . 218.034 Midterm #2 Name: 3. (a) (10 points) If the constant a is not a root of the polynomial p with real coefficients, show that a particular solution of p(D)y = eat is atey(t) = . p(a)(b) (10 points) Find a particular solution of (D4 + D3 + D2 + D + 1)y = 33e−2t . 318.034 Midterm #2 Name: 4. (20 points) Find the general solution of y��� + 3y�� + 3y� + y = te−t . 418.034 Midterm #2 Name: 5. Consider the initial value problem y� = 1 + y 2 , y(0) = 0. (a) (10 points) Using Picard’s iteration method obtain the iterates y1(t) and y2(t). (b) (10 points) Show that the initial value problem has at most one solution in any interval of the form t ∈ (−a, a). (c) (extra credits) Find the exact solution y(t) and show that limn→∞ yn(t) = y(t).
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