q_a_16

110321

Question:

`q001.  Write out the system of two equations corresponding to each of the following matrix equations:

[ 2, 3 ; -2, 1 ] * [y_1, y_2] ` = [y_1' , y_2' ]`

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[ t, sin(t); 1/t, t^2 ] * [y_1, y_2] ` = [y_1' , y_2' ]`

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[ 2, 3 ; -2, 1 ] * [y_1, y_2] ` + [1, t] ` = [y_1' , y_2' ]`

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Question:

`q002.  Write each of the systems in matrix form:

system 1:

y_1 ' = 3 y_1 + 2 y_2

y_2 ' = 2 y_1 - 4 y_2

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system 2:

y_1 ' = 2 t y_1 - cos(t) y_2 + 4

y_2 ' = 3 y_1 - 4 / t y_2 + 1

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Question:

`q003.  Find the derivative of the matrix

[ t, sin(t); 1/t, t^2 ]

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Find an antiderivative of this same matrix.

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Question:

`q004.  Let A = [ t, sin(t); 1/t, t^2 ] and B = [ cos(t), t; sqrt(t), 1/t ].

Show that (A B) ' = A ' B + A B '.

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Question:

`q005.  Find the value(s) lambda for which A x = lambda x, where A is the matrix

[ 2, 3 ; 2, 1 ]

and x is an unspecified vector.

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Question:

`q006.  For each value of lambda found in the preceding, find a vector x for which the equation A x = lambda x is true.

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7.  Let lambda_1 and lambda_2 be the solutions you found for #5, and x_1, x_2 the corresponding vectors you found in #6.

Show that y = [y_1, y_2] ` = e^(lambda_1 * t) * x_1 is a solution to the equation

[ 2, 3 ; 2, 1 ] [y_1, y_2] `  = [y_1 ' , y_2 ' ] `

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Show the analogous result for e^(lambda_2 * t) * x_2.

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Show furthermore that

c_1 e^(lambda_1 * t) * x_1 + c_2 e^(lambda_2 * t) * x_2

is a solution to the equation.

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