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Unformatted text preview: GE 207K 2nd Order ODEs with Cons. Coeffs PS September 22, 2010 Problem 1 Find the general solution to the following differential equation the general solution to the following ODE: y 00 + 5 y + 6 y = 0 . (1) Solution : The corresponding characteristic equation is r 2 + 5 r + 6 = 0 , which can be factored as ( r + 3) ( r + 2) = 0 . Therefore, the roots real and distinct, given by r 1 = 3 , r 2 = 2 . Therefore, the general solution is y = c 1 e 3 t + c 2 e 2 t . (2) 1 GE 207K 2nd Order ODEs with Cons. Coeffs PS September 22, 2010 Problem 2 Solve the initial value problem (IVP) 6 y 00 + 5 y 6 y = 0 , y (0) = 0 ,y (0) = 1 . (3) Solution : The corresponding characteristic equation is 6 r 2 + 5 r 6 = 0 . Rewrite the characteristic equation (3 r 2) (2 r + 3) = 0 . The roots of the equation are real and distinct, given by r 1 = 2 3 , r 2 = 3 2 . Therefore, the general solution to the differential equation is given by y = c 1 e 3 / 2 t + c 2 e 2 / 3 t . Were given intial conditions, and therefore we can solve for the two arbitrary constants. y (0) = 0 = c 1 + c 2 , y (0) = 1 = 3 2 c 1 + 2 3 c 2 , c 2 = c 1 = 6 13 ....
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 Fall '10
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