Consider the second order differential equation y 00

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Consider the second order differential equation y 00 - 4 y 0 + 4 y = 2 e 3 x . (1) (a) Find the general solution of the homogeneous equation corresponding to (1). (b) Find a particular solution of the inhomogeneous equation (1). (c) Solve the initial value problem given by (1) and initial conditions y (0) = 0, y 0 (0) = - 1.
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EGR 265, Spring 2012, Final Exam 5 Problem 4 (12 points) A mass of 100 kg stretches an undamped spring by 10 cm. Assume that g = 10 m/s 2 . Include the correct units in all your answers below. (a) Find the spring constant k and its correct unit. (b) Set up the second order differential equation which governs the motion of the spring-mass system, choosing the x -axis to be oriented downwards. Find the general solution of this equation. (c) Find the particular solution of the equation if the mass is released 50 cm below the equilibrium position from rest. (d) What is the first positive time at which the mass returns to the equilibrium position?
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EGR 265, Spring 2012, Final Exam 6 Problem 5 (10 points) (a) Find the gradient of f ( x, y ) = ye xy . (b) Evaluate the directional derivative of f ( x, y ) at the point with coordinates (0 , 1) in the direction of the vector from (0 , 1) to (1 , 3). (c) Find a unit vector in the direction of steepest increase of f ( x, y ) at the point (0 , 1).
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EGR 265, Spring 2012, Final Exam 7 Problem 6 (10 points) (a) Determine the equation of the tangent plane to the graph of z = x x + y through the
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