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' = 8x+85y y'= -x-4y y'= -x-4y ==> x=- y'-4y ==> x' = -y' '- 4y' x' = 8x+85y -y'' -4y' = 8(-y'-4y) + 85 y -y'' - 4 y' = -8 y' -32y +85 y -y' ' +4 y ' = 53 y y'' - 4y' + 53y =0 m^2-4m + 53=0 m=2- 7i m=2 + 7i y(t) = e^(2t)( C1 cos( 7 t) + C2 sin( 7 t) ) y' = 2 e^(9t)( C1 cos( 7 t) + C2 sin( 7 t) )+ e^(2t)( - 7 C1 sin( 7 t) + 7 C2 cos( 7 t) ) x= - y'-4y ==> x(t) = -( 2 e^(9t)( C1 cos( 7 t) + C2 sin( 7 t) )+ e^(2t)( - 7 C1 sin( 7 t) + 7 C2 cos( 7 t) ) ) - 4( e^(2t)( C1 cos( 7 t) + C2 sin( 7 t) )) x(t) = e^(2 t) (-(7 C2 + 2 C1 (2 + e^(7 t))) Cos[ 7 t] + (7 C1 - 2 C2 (2 + e^(7 t))) Sin[7 t]) Thus x(t) = e^(2 t) (-(7 C2 + 2 C1 (2 + e^(7 t))) cos( 7 t) + (7 C1 - 2 C2 (2 + e^(7 t))) sin(7 t) ) y(t) = e^(2t)( C1 cos( 7 t) + C2 sin( 7 t) )

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