{x\' = - x - 2y y\' = 8x - y Solution x\'=-x-2y ==> 2y= -x\'-x ==> y= -(x\'+x)/2

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x'=-x-2y ==> 2y= -x'-x ==> y= -(x'+x)/2 ==> y'=-(x''+x')/2 y'= 8x-y ==> -(x''+x')/2 = 8x - (-(x'+x)/2 ) -(x''+x') = 16x + (x'+x) -x'' -x' = 16x + x'+x x'' +2x'' +17 =0 m^2 +2m+17=0 m=-1 -4i OR m=-1+4i x(t) = C1 exp((-1-4i)t) + C2 exp((-1+4i)t) x(t) = exp(-t) ( A cos(4t) + B sin(4t) ) x'= exp(-t) ( 4 B cos(4 t) - 4 A sin(4 t) ) - exp(-t) (A cos(4 t) + B sin(4 t) ) y= -(x'+x)/2 ==> y(t) = -(x'+x)/2 y(t)=- ( exp(-t) ( 4 B cos(4 t) - 4 A sin(4 t) ) - exp(-t) (A cos(4 t) + B sin(4 t) ) +exp(-t) ( A cos(4t) + B sin(4t) ) )/2 y(t) = - ( exp(-t) (4 B Cos[4 t] - 4 A Sin[4 t]) /2

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