q_a_20

110404 Differential Equations

1.  Express the equation

y '''' + 3 y ''' - 2 y '' + y ' - y = 0

as a system of equations in the variables u_1 = y, u_2 = y ', u_3 = y '', u_4 = y ''' .

Start by letting u_vector = [u_1; u_2; u_3; u_4] and write its derivative u_vector ', using the fact that u_4 ' = y '''' = -(3 y ''' - 2 y '' + y ' - y).

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2.  Two tanks with equal volumes V each have an input and an output pipe.  In addition two pipes connect the two tanks, with water in the two pipes flowing in opposite directions.  The flow rate in every pipe in the system is r.  The amount of solute in tanks 1 and 2 can be denoted Q_1 and Q_2.

Find the matrix A such that [Q_1; Q_2] ' = A * [Q_1; Q_2].

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Find the general solution to the equation.

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