y\" = -y Assume that the function y = y(t) is measured in centimeters and t is m
ID: 2873736 • Letter: Y
Question
y" = -y Assume that the function y = y(t) is measured in centimeters and t is measured in seconds. In this case a solution to the differential equation can be thought of as the motion of an object connected to a spring. Show that y(t) = A cos t + B sin t is a solution to the differential equation where A and B are any fixed real constants. Consider the initial condition y(0) = 1. In words, we could describe this physically as "At time 0 the object is located 1 cm to the right." Interpret the following conditions in single sentences: y(0) = 0, y(1) = -.5, y'(0) = 2. y'(2) = -1. Write down mathematical conditions for the following statement: "At time 0 I draw the object 2 centimeters to the left. One second later, it is moving 4 cm/s to the right." Give an exact solution to the differential equation (1) under the following conditions: At time 0 the object is located at the origin and has velocity 3 cm/s to the right. Give an exact solution to the differential equation (1) under the condition that at time zero, you draw the object 2 cm to the left and then let it go.Explanation / Answer
(a)
Consider the given differential equation:
To show that , find the its second derivative.
Therefore, is solution for the differential equation .
(b)
: At time 0 the object not moved anywhere from the initial position.
: At time 1 the object is located to the left.
: At time 0 the object moving 2 cm/sec to the right
: At time 2 the object moving 1 cm/sec to the left.
(c)
As at time 0, the object 2 cm to the left, means .
After one second, it moving 4 cm/s to the right, thus means .
(d)
Construct the mathematical conditions for the statements “ At the time 0 the object is located at the origin and has velocity 3 cm/sec to the right”
That is,
Substitute the obtained conditions in and .
Therefore, the solution of the differential equation at the given conditions is,
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