110411 Differential Equations
Question:
`q001. Find solutions of
y ' = [ 5, 3; -4, -3 ] y
for which y(1) = [2; 0].
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Find the expression for W(t), the Wronskian of the solution.
Show that W(t) = W(1) * e^(integral(2 ds, s from 1 to t).
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Question:
`q002. We have seen (and should be able to prove) that the Laplace transform of the derivative function of the function y(t) is given by
L(y ' ) = s * L(y) - y(0)
You should have done the integral to show that
L(e^(alpha t)) = 1 / (s - alpha)
(the integral is just integral( e^(alpha t) * e^(-s t) dt, t from 0 to infinity) ).
We can use these facts to solve the familiar equation
y ' = - k y
Transform the equation and solve for L(y).
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Having found L(y), find the function of which L(y) is the transform.
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Question:
`q003. The Heaviside function is
H(t) =
0 if t < 0,
1 if t >= 0.
What is the Laplace Transform of H(t)?
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Question:
`q004. What is the definition of the function H(t - alpha)?
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What is the Laplace Transform of H(t - alpha)?
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Question:
`q005. Sketch a graph of the following function, and give a description of your graph and the process you used to construct it:
H(t) - 2 H(t - 1) + 2 H(t - 2) - 2 H(t - 3).
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What is the Laplace Transform of this function?
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What does this function have to do with the square waves you have been generating in physics class?
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Question:
`q006. What is the Laplace Transform of the function y = A cos(omega t + phi)?
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What might this have to do with alternating circuits?
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Question:
`q007. See if you can figure out the formula for L(y ''). Hint: Two easy integrations by parts will do it. Just write down the integral and go to it.
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