Sample Final for math 251 PDF

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Summary

This is a practice exam for math 251 final and has 11 questions...

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Name: MATH 251 Fall 2021

Student Number: Instructor: Section:

Sample Final

This exam has 11 questions for a total of 150 points.

You must show work for full credit. Only Real Valued solution are accepted.

Notes, devices, communication with other people is NOT permitted (except to ask an instructor a question).

1. (10 points) Solve the IVP: 2ty y + 2 = t +4 ′

r

t3 + 4t 3

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y(0) =

1 4

2. (10 points) Solve the IVP y′ = y2 cos(t)

y(0) = −π

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3. (10 points) Solve the IVP: 3 ′

4y y −

√ t

e

2

=0

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y(0) = −2

4. (15 points) The position (measured in meters) of a mass tied to a spring hanging from the ceiling is modeled by the IVP: u′′ + 7u′ + 12u = 0

u(0) = 5, u′ (0) = 0

in which positive values of u represent positions below the equilibrium position (u = 0). Assume the mass begins moving at time t = 0. a) Find the position of the mass u(t) for all time t ≥ 0

b) List all the times the mass passes through the equilibrium position (if ever) and label each time with the direction of the mass. You must show work.

c) At what time is the mass at its lowest point?

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5. (20 points) Compute

a)

L−1



e−2s (s − 1)(s + 1)



b) L{(t2 − 6)u6 − δ(t − 6)}

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6. (10 points) Find the general solution to the linear homogeneous system , x′ = Ax where   1 5 A= −5 −9

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7. (15 points) Solve the linear homogeneous system with given initial condition, x′ = Ax where   0 −2 A= 2 0 and x(0) =



1 12



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8. (10 points) For the function below, assume it is periodically extended outside the original interval.    1 −4 ≤ x < 0 f (x) = 0 x=0   −1 0...