See also Class Notes 110223
Question:
`q001. Using the substitution y = A e^(r t) find two linearly independent solutions to the equation y '' + y ' - 6 y = 0.
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Show that your solutions are linearly independent.
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Form two different linear combinations of your solutions and show whether they are linearly independent.
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Question: `q002. Consider the equation y '' + sqrt(t) y ' + 1 / (t^2 - 4) y = 1 / cos(t).
For what, if any, values of t is the equation undefined?
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What is the largest region of the y vs. t plane which contains the point (t_0, y_0) for each of the following pairs of values?
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Question:
`q003. What is a possible function y_2(t) such that the Wronskian of y_1(t) = e^(3 t) and y_2(t) is e^t?
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Question: `q004. Suppose that y_1 and y_2 are linearly independent functions of t, and that we wish to find a solution of the form y(t) = c_1 * y_1(t) + c_2 * y_2(t) such that y(t_0) = y_0 and y ' (t_0) = v_0. The values y_0 and t_0 are considered to be given, and we want to solve for c_1 and c_2.
Write out the two equations that correspond to our given conditions y(t_0) = y_0 and y ' (t_0) = v_0.
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Write this system as a matrix equation.
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What does this equation have to do with the Wronskian?
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Question: `q005. Suppose that y_1(1) = 2, y_2(1) = 4, (y_1) ' (1) = 5 and (y_2) ' (1) = 3. What therefore is the value of W(1), where W(t) is the Wronskian of the set (y_1(t), y_2(t))?
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Give a set of values for y_1(1), y_2(1), (y_1) ' (1) and (y_2) ' (1) that would make the Wronskian zero.
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`q006. Suppose that y(t) = u(t) * e^(-alpha * t).
Write the equation y '' + 2 alpha y ' + alpha^2 y = 0 for this function.
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If we know that y_1 = e^(-alpha t) is a solution to the equation y '' + 2 alpha y ' + alpha^2 y = 0, how does the equation you have obtained simplify?
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What does this result tell us about the function u?
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What do we conclude about our function y = u(t) * y_1(t)?
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If our new solution denoted is y_2(t), then show that { e^(alpha t), (c t + d) e^(alpha t)} is a fundamental set.
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Question:
`q007. The net force on a simple pendulum of mass m at position x is F_net = - k x - gamma x '. - k x is the restoring force (from physics we know that k = m g / L, and -gamma x ' is the drag force that results from air resistance).
We know that F_net = m * a = m * x ''.
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What therefore is the differential equation equivalent to the given force equation?
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Using trial solution x = e^(r t) find two linearly independent solutions of this equation, and show that they are linearly independent.
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