Question

Differential Equation class. Show the steps for the solutions please.

Section 3.2 Exercises To Solutions 1. Suppose that the mass is set in motion by moving it upward by 2.5 cm and releasing it with no initial velocity (a) Sketch what you think the graph of y versus t will look like, taking care with the fact that positive y is upward. Make the amplitude of the motion clear on your graph. (b) Express the initial conditions mathematically by giving values for y(0) and y(0). 2. Repeat Exercise 1 for a mass that is set in motion by hitting it upward (as it hangs at equilibrium), giving it an initial speed of 3 inches per second. 3. Repeat Exercise 1 for a mass that is set in motion by giving it an initial speed of 8 cm per second downward from a point 2 cm above equilibrium. The amplitude is not two - make it clear whether t is more or less than two. 4. A mass ofis attached to a spring with a spring constant of 15. raising it 2.5 cm and releasing it. (See Exercise 1.) It is set into motion by (a) Set up an initial value problem consisting of a differential equation and two initial conditions. (b) Solve the initial value problem to find the position function y. Give all numbers as decimals, rounded to the hundredth's place. (c) Graph your solution using some technology and compare with your answer to Exercise 1. They should agree; if they don't, figure out what is wrong and fix it! (d) Put your solution in the form Csinfwt +. Graph the resulting function and make sure the graph agrees with what you got for (c). If not, try to find and correct your error. (e) Give the amplitude, angular frequency, period, frequency and phase shift of the solution. 5. (a) Consider again a mass of attached to a spring with a spring constant of 15, as in Exercise 4. Solve the initial value problem obtained with the initial conditions of Exercise 3, where the initial speed was 8 cm downward from a point 2 cm above equilibrium. Again giv numbers as decimals, rounded to the hundredth's place. (b) Graph your solution using technology and compare with your answer to Exercise 3. (c) Put your solution in the form yAsin(ut). Check it by graphing it and your solution to part (a) together; they should be the same! (d) Give the amplitude, angular frequency, period, frequency and phase shift of the solution. 6. A mass of on a spring with spring constant 4 is given an initial velocity of 9 cm/sec upward from an initial position of 4 cm below equilibrium. (a) Give the initial value problem. (b) Find the equation of motion of the mass, y(t), in the form y = Asin(wt + φ). (c) Give the amplitude, angular frequency, period, frequency and phase shift of the solution

Answer 1

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