Find the general solution of the following homogeneous system of linear equation

Question

Find the general solution of the following homogeneous system of linear equations. NOTE: For answers of the form e lambda t, enter the components of the vector in a column and enter the function e lambda t in the following answer box.

Explanation / Answer

x' = 9x+12y y'= -3x+9y x' = 9x+12y==> 12y= x'-9x ==> y= (x'-9x)/12 ==> y'= (x' '- 9x' ) / 12 y'= -3x+9y (x' '-9x')/12 = -3x + 9 (x'-9x)/12 x'' - 9x' = -36 x + 9x' - 81x x' ' - 18x' = -117x x'' - 18x' + 117x =0 m^2-18m + 117=0 m=9- 6i m=9 + 6i x(t) = e^(9t)( C1 cos( 6 t) + C2 sin( 6 t) ) x' = 9 e^(9t)( C1 cos( 6 t) + C2 sin( 6 t) )+ e^(9t)( - 6 C1 sin( 6 t) + 6 C2 cos( 6 t) ) y= (x'-9x)/12 ==> y(t) = ( 9 e^(9t)( C1 cos( 6 t) + C2 sin( 6 t) )+ e^(9t)( - 6 C1 sin( 6 t) + 6 C2 cos( 6 t) ) - 9 ( e^(9t)( C1 cos( 6 t) + C2 sin( 6 t) ) ) ) / 12 y(t) = e^(9t)( C1 cos( 6 t) -(1/2) C2 sin( 6 t) ) Thus x(t) = e^(9t)( C1 cos( 6 t) + C2 sin( 6 t) ) y(t) = e^(9t)( C1 cos( 6 t) -(1/2) C2 sin( 6 t) )

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