Find general solution for the ODE y"-9y=(9x)/(e^(3x)) Write clean, and clear. Show steps of calculations. Hint: use variation of parameters method for finding particular solution y_p

Tamara Bryan 2022-07-22 Answered
Find general solution for the ODE
y " 9 y = 9 x e 3 x
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Answers (1)

Mira Spears
Answered 2022-07-23 Author has 14 answers
y " 9 y = 9 x e 3 x
Auxiliary eqn is m 2 9 = 0
or  m 2 = 9 m = ± 3 y n ( x ) = c 1 e 3 x + c 2 e 3 x let  y 1 = e 3 x , y 2 = e 3 x , r ( x ) = 9 x e 3 x = 9 x e 3 x w ( y 1 , y 2 ) = | e 3 x e 3 x 3 e 3 x 3 e 3 x | w ( x ) = 3 e 0 3 e 0 = 3 3 = 6
using variation of parameters,
let  y p = A y 1 + B y 2  be any particular solution where  A = y 2 r ( x ) w ( x ) d x = e 3 x a x e 3 x 6 d x A = 3 2 x e 6 x d x A = 3 2 [ x e 6 x 6 e 6 x 36 ] = e 6 x ( x 4 1 24 ) B = y 1 r ( x ) w ( x ) d x = e 3 x a x e 3 x 6 d x = 9 6 x d x B = 3 2 x 2 2 = 3 4 x 2 y p = e 3 x e 6 x ( x 4 1 24 ) 3 4 e 3 x x 2 = e 3 x ( x 4 1 24 3 4 x 2 )
General solution y ( x ) = y n ( x ) + y p ( x ) y ( x ) = c 1 e 3 x + c 2 e 3 x + e 3 x ( x 4 1 24 3 4 x 2 ) let  c 2 1 24 = k 2 y ( x ) = c 1 e 3 x + k 2 e 3 x + e 3 x ( x 4 3 4 x 2 )

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