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 Problem Set #7 (modified on April 13, 2009) Due by Friday, April 17, 2009, by NOON 1. In many applications the input f and the output y are related by L2y = L1f, where L1 and L2 are linear differential operators. (a) If Pj is the characteristic polynomial of Lj , j = 1, 2, show that P2(s)Y (s) = P1(s)F (s) + P0(s), where P0(s)is a polynomial depending on the initial conditions of both f and y. (b) The null initial condition gives the transfer function W (s) = P1(s)/P2(s). If W (s) = Lw, express y by the convolution theorem. (c) We can have a function w as in part (b) only if the degree of P1 is less than that of P2. Why? What happens to the formula of part (b) if P1 = P2? 2. One method of determining the ratio e/m of charge to mass for an electron leads to the system mx�� + Hey� = eE, my�� − Hex� = 0. The initial values are x(0) = x�(0) = y(0) = y�(0) = 0 and m, e, H, E are positive constants. (a) Solve by the Laplace transform and show that the path of a particle in the coordinates (x, y) are given by x(t) = E(ωH)−1(1 − cos ωt), y(t) = E(ωH)−1(ωt − sin ωt), where ω = He/m. This is a cycloid generated by a circle of radius E(ωH)−1 . (b) How can one determine the system-parameter e/m from knowledge of the path and the values E and H set by the experimenter? 3. Consider the system of differential equations Y � = AY , where y11(t) y12(t) a11(t) a12(t)Y = , A = . y21(t) y22(t) a21(t) a22(t) (a) If U(t) is a fundamental solution and C is a nonsingular matrix, show that U C is also a funda-mental solution. (b) If V (t) is a fundamental solution, show that U(t) = V (t)V (t0)−1 satisfies U(t0) = I and is also a fundamental solution. (c) Show that | Y � = D1 + D2, where | D1 = a11y11 + a12y21 a11y12 + a12y22 y21 y22 = a11|Y | and D2 = a22|Y |. Deduce the Liouville theorem. 4. (a) Show that if (y1(t), y2(t)) is a solution of t2 y�= −2ty1 + 4y2, ty�= −2ty1 + 5y21 2 then both y1 and y2 are solutions of the Euler equation (t2φ� + 2tφ)� = −8φ + 5(tφ� + 2φ). 1 .� � � � (b) Find a fundamental matrix for −2t−1 4t−2 −2 5t−1 (c) For A =1 2t −2 obtain a fundamental matrix. t2 t2 + 2t −2 5. Solve the initial value problem � � � �� � � � � � dy1 1 4 y1 y1(0) 0 dt y2 =2 −1 y2 ,y2(0) = 3 (a) by using the eigenvalues and (b) by using the Laplace transform. (c) For which values of a and b, the solution of the initial value problem � � � �� � � � � � dy1 1 4 y1 y1(0) a dt y2 = −1 y2 , = 2 y2(0) b has the behavior limt→∞(y1(t), y2(t))T = (0, 0)? 6. For which vector (a, b, c), is the solution of the initial value problem ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y1 1 0 −2 y1 y1(0) ad ⎝y2⎠ = ⎝0 1 0 ⎠⎝y2⎠ , ⎝y2(0)⎠ = ⎝b⎠ dt y3 1 −1 −1 y3 y3(0) c periodic in t?
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