solutions4ac - Math 33b Quiz 4ac February 5 2008 Name UCLA...

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Math 33b, Quiz 4ac, February 5, 2008 Name: UCLA ID: 1. Find the general solution to the differential equation y 00 - 3 y 0 - 10 y = e - 2 t . Solution. The general solution y ( t ) is of the form y = y h + y p , where y h is a solution to the homogeneous equation y 00 - 3 y 0 - 10 y = 0 and y p is one particular solution to the inhomogeneous equation. The characteristic polynomial for the homogeneous equation is λ 2 - 3 λ - 10 = 0, which has roots λ = - 2 , 5. Hence the homogeneous solution has the form y h ( t ) = C 1 e - 2 t = C 2 e 5 t . For the particular solution y p , the right-hand side of the differential equation suggests trying a function of the form
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Unformatted text preview: p = Ae-2 t , but this term is already included as part of the homogeneous solution. Hence we try a solution of the form y p = Ate-2 t . We thus want to find A such that y 00 p-3 y p-10 y p = e-2 t . We compute y p = Ae-2 t-2 Ate-2 t and y 00 p =-4 Ae-2 t + 4 Ate-2 t . Crunching everything through, we get (-4 Ae-2 t + 4 At 3-2 t ) + (-3 Ae-2 t + 6 Ate-2 t ) + (-10 Ate-2 t ) = e-2 t-7 Ae-2 t = e-2 t A =-1 7 Hence y = y h + y p = C 1 e-2 t + C 2 e 5 t-1 7 te-2 t . 1...
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