Then the general solution of the homogeneous DE ay 00 by cy 0 satisfies y t c 1

Then the general solution of the homogeneous de ay 00

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Then the general solution of the homogeneous DE , ay 00 + by 0 + cy = 0 , satisfies y ( t ) = c 1 e λ 1 t + c 2 e λ 2 t if λ 1 6 = λ 2 are real, y ( t ) = c 1 e λ 1 t + c 2 te λ 1 t if λ 1 = λ 2 , y ( t ) = c 1 e μt cos( νt ) + c 2 e μt sin( νt ) if λ 1 , 2 = μ ± are complex. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (14/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Homogeneous Equations - Example Consider the IVP y 00 + 5 y 0 + 6 y = 0 , y (0) = 2 , y 0 (0) = 3 . The characteristic equation is λ 2 +5 λ +6 = ( λ +3)( λ +2) = 0, so λ = - 3 and λ = - 2 The general solution is y ( t ) = c 1 e - 3 t + c 2 e - 2 t From the initial conditions y (0) = c 1 + c 2 = 2 and y (0) = - 3 c 1 - 2 c 2 = 3 When solved simultaneously, gives c 1 = - 7 and c 2 = 9, so y ( t ) = 9 e - 2 t - 7 e - 3 t This problem is the same as solving ˙ x = 0 1 - 6 - 5 x , x (0) = 2 3 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (15/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Nonhomogeneous Equations 1 Nonhomogeneous Equations: Consider the DE L [ y ] = y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) Theorem Let y 1 and y 2 form a fundamental set of solutions to the homogeneous equation , L [ y ] = 0 . Also, assume that Y p is a particular solution to L [ Y p ] = g ( t ) . Then the general solution to L [ Y ] = g ( t ) is given by: y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + Y p ( t ) . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (16/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Nonhomogeneous Equations 2 The previous theorem provides the basic solution strategy for 2 nd order nonhomogeneous differential equations Find the general solution c 1 y 1 ( t ) + c 2 y 2 ( t ) of the homogeneous equation This is sometimes called the complementary solution and often denoted y c ( t ) or y h ( t ) Find any solution of the nonhomogeneous DE This is usually called the particular solution and often denoted y p ( t ) Add these solutions together for the general solution Two common methods for obtaining the particular solution For common specific functions and constant coefficients for the DE, use the method of undetermined coefficients More general method uses method of variation of parameters Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (17/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Method of Undetermined Coefficients Method of Undetermined Coefficients - Example 1: Consider the DE y 00 - 3 y 0 - 4 y = 3 e 2 t The characteristic equation is λ 2 - 3 λ - 4 = ( λ + 1)( λ - 4) = 0, so the homogeneous solution is y c ( t ) = c 1 e - t + c 2 e 4 t Neither solution matches the forcing function , so try y p ( t ) = Ae 2 t It follows that 4 Ae 2 t - 6 Ae 2 t - 4 Ae 2 t = - 6 Ae 2 t = 3 e 2 t or A = - 1 2 The solution combines these to obtain y ( t ) = c
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