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m238sum08t3h - Mathematics 238 test three due Monday August...

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Mathematics 238 test three due Monday, August 4, 2008 please show your work to get full credit 1 . Given the system dx dt = - x + 6 y dy dt = - 6 x - y ( a ) Find the real-valued general solution to the system. ( b ) Sketch the phase plane ( xy -plane) for the system, showing the direction field, the iso- clines, and a few representative solutions. ( c ) Solve the differential equation for solutions in the phase plane dy dx = - 6 x - y - x + 6 y 2 . Find the real-valued general solution of d 2 x dt 2 = y - x - 1 d 2 y dt 2 = - ( y - x - 1) + ( z - y - 1) d 2 z dt 2 = - ( z - y - 1) 3 . Solve the differential equation d 2 y dx 2 - x dy dx +2 y = 0 using power series centered at the origin ( a ) For what values of x will the power series solutions converge? ( b ) Find the recurrence relation for the power series ( c ) Write down the general solution through degree 5 terms ( d ) Solve the initial value problem y (0) = 1 and y (0) = 0. Write down the solution through
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