Question

dy =1 +1 dt X-- Solve the system: 5.) dx dy 2y=e + dt d

Can anyone help me with this Linear Algebra question?

Answer 1

"x - d^2 y/dt^2 = t + 1 dx /dt + dy/dt - 2y = e^t Differentiate a w r t t dx/dt - d^3 y/dt^3 = 1 from b dx/dt = e^t - dy/dt + 2y substitute d in c e^t - dy/dt + 2y - y""' = 1 implies - y""' - y' + 2y = 1 - e^t implies y""' + y' - 2y = e^t - 1 homogenous D.E y""' + y' - 2y = 0 Auxiliary eqn m^3 + m - 2 = 0 implies (m - 1) (m^2 + m + 2) m = 1, m = -1 plusminus squareroot 1 - 8/2 = -1 plusminus squareroot 7 i/2 so general solution of homogenous eqn y_n(t) = c_1 e^t + (x_2 cos(squareroot 7 t/2) + c_3 sin (squareroot 7/2 t)) e^- t/2 now finding particular integral (D^3 + D - 2) = e^t - 1 implies y_p = 1/D^3 + D - 2 (e^t - 1) = 1/(D - 1) (D^2 + D + 2) (e^t - 1) = 1/D - 1[1/D^2 + D + 2 e^t] - 1/D^3 + D -2 e^0.t = 1/D - 1 "