100223 Differential Equations

Consider the following:

• We can solve first-order linear, exact and separable equations.
• We can reduce a Bernoulli equation to a first-order linear equation.
• We can solve the logistic equation, which is separable, and related models.
• We can solve problems involving air resistance and Newton's Law of Cooling.
• We can solve motion problems where position, rather than time, is the independent variable.
• We can use direction fields and the Euler method.
• We are therefore, or soon will be, ready for a test on Chapters 1 - 3.

You should work through the assigned problems for Chapters 2-3 within the next week.  The problem numbers might be fore a different edition of the text, so these problems are posted as qa's (#'s 5 - 11).

Assigned problems for Chapter 4:

Introductory section problems 1 and 2 (modified to metric units in query 12)

Section 4.1 problems 2, 4, 11 (query 13)

Section 4.2 problems 2, 12, 22 (query 14)

Section 4.3 problems 4, 8, 12, 20 , 26 (query 15)

Section 4.4 problems 2, 10, 18, 22 (query 16)

Section 4.5 problems 4, 10, 16 (query 17)

Section 4.6 problems 6, 12, 16, 20, 36 (query 18)

Section 4.7 problems 2, 6, 10 (query 19)

Section 4.8 problems ... (problems for 4.8 - 4.11 will be posted later)

`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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`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?

• t_0 = pi, y_0 = 1
• t_0 = 0, y_0 = -1
• t_0 = 1, y _0 = 0

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`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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`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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`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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`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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