Review · Aristotle

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Why must the coefficient of

y^{'}

be 1 before reading off

p (t)

?

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Which function solves

y^{'} + 2 y = 6

and

y (0) = 1

?

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For

y^{'} = y (1 - y)

, classify 0 and 1 using the signs immediately on either side.

Your answer

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Match each logistic initial state to its forward behavior on its interval of existence.

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· Exam style · 6 pointsFlag for review

Solve an IVP and verify the initial data

Solve

y^{'} + 4 y = 8

,

y (0) = 5

, and verify the initial value and long-time limit.

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· Exam style · 3 pointsFlag for review

Separate equilibrium behavior from transient behavior

For

y^{'} = y (2 - y)

, which claim is correct?

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Match each root structure to a real fundamental pair.

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For

y = C_{1} e^{- t} + C_{2} e^{- 3 t}

, write the two equations imposed by

y (0) = 2

and

y^{'} (0) = 0

.

Your answer

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For

m = 1

,

k = 4

, which damping coefficient is critical?

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For

E = \frac{1}{2} m (x^{'})^{2} + \frac{1}{2} k x^{2}

, show that a free damped oscillator cannot gain mechanical energy.

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· Exam style · 8 pointsFlag for review

Solve an IVP and verify the initial data

A unit mass obeys

x^{''} + 6 x^{'} + 13 x = 0

,

x (0) = 2

,

x^{'} (0) = 0

. Find the displacement, classify damping and give the damped angular frequency.

Your answer

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If

y (0) = 4

, what is

L {y^{'}}

?

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Why should the original IVP be checked after inverse transformation?

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Invert

e^{- 3 s} / (s + 2)

, and state when the result is zero.

Your answer

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Match each eigenvalue pair to its planar linear classification, assuming the stated real parts are nonzero.

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· Exam style · 6 pointsFlag for review

Connect eigenvalues, eigenvectors and trajectory behavior

Match each item for

A = diag (- 1, - 2)

to its meaning.

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· Exam style · 6 pointsFlag for review

Carry initial terms and time shifts through the transform

Invert

Y (s) = e^{- 2 s} / [s (s + 3)]

and check the differential equation

y^{'} + 3 y = u (t - 2)

away from the switch.

Your answer

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