dx/dt= -.2y dy/dt= .8x - 0.1 y find x(t) and y(t) and y(x) given x(0) =2 and y(0

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dx/dt= -0.2y ==> x' = -0.2 y ==> -0.2 y' = x'' ==> y' = - x'' / 0.2 dy/dt= 0.8x - 0.1 y ==> y'= 0.8 x- 0.1 y ==> - x'' / 0.2 = 0.8x - 0.1 ( x' / 0.2 ) ==> -x'' = 0.16 x - 0.1 x' ==> x'' -0.1 x' + 0.16x=0 m^2 - (1/10) *m + (16/10) == 0 roots are m1 = 0.05 - 1.26392 i and m2= 0.05 + 1.26392 i x(t) = C1 e^(m1 t) + C2 e^(m2 t) ==> x'= m1 C1 e^(m1 t) + m2 C2 e^(m2 t) x' = -0.2 y ==> y= - x' / 0.2 = -5 x' y(t) = -5 (m1 C1 e^(m1 t) + m2 C2 e^(m2 t)) find x(t) and y(t) and y(x) given x(0) =2 and y(0)=0 x(t) = C1 e^(m1 t) + C2 e^(m2 t) ==> x(0)=2=> C1+ C2 = 2 y(t) = -5 (m1 C1 e^(m1 t) + m2 C2 e^(m2 t)) ==> y(0)=0==> -5C1m1 -5C2m2 =0 C1+ C2 = 2 -5C1m1 -5C2m2 =0 Solve[{C1 + C2 == 2, -5*C1*m1 - 5*C2*m2 == 0}, {C1, C2}] C1 = -2m2 / (m1 - m2) C2 = 2m1 / (m1 - m2)

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