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Exam II Review 1 Qualitative Behavior Consider the following 2nd order ODE’s (a) y 00 + 2y 0 + y = 0 √ (b) 3y 00 + 2y 0 + y = 0 (c) y 00 + 1.1y = cos(t) (d) y 00 + 4y = sin(2t + π/2) Without solving, do the following for each equation given above: name the phenomena exhibited by the equation and give justification. Finally, sketch the qualitative behavior of the solution. 2 Solution Techniques For each of the equations listed below, state which method is the most appropriate solution technique (give justification). Then, solve the equation. (a) y 00 − y 0 − 2y = et , y(0) = y 0 (0) = 1 R (b) y 00 + y = sec2 (x), Hint: sec(x)dx = ln | sec(x) + tan(x)| + c (c) y 00 + y = 1 + U t − π2 [sin(t) − 1] , y(0) = 1, y 0 (0) = 0 3 Spring-Mass System A force of 400 Newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and it is initially released from the equilibrium position with an upward velocity of 10 m/s. Additionally, the mass is driven by an external force equal to f (t) = sin(2t). (a) Write the corresponding IVP. (b) Solve the IVP using any appropriate method. Name the behavior of the solution. 1 4 Laplace Transform (a) For the following equation, find the Laplace transform using only the definition: f (t) = et cos(t) (b) Find the Laplace transforms of the following functions using the table: (i) g(t) = (t + 1)3 0 t < π/2 (ii) h(t) = cos(3(t − π/2)) t ≥ π/2 5 Inverse Laplace Transform Find the inverse Laplace transform of the following functions: (a) Y (s) = 2s+5 s2 +6s+34 (b) Q(s) = e−s (2s−1) s2 (s+1)2 6 One More Thing There are a few topics that are not included on this review, but that doesn’t mean they won’t be on the exam. You are still responsible for all material covered in lecture! 2