Question

Solve the given system of differential equations by systematic elimination dx 20y dt dy = X + Z dt dz = X + y dt (x(t), y(t), z(t))

Answer 1

The given system is dx/dt = 20 y dy/dt = x + z dz/dt = x + y By (ii) - (iii) d/dt (y - z) = -(y - z) = > d(y -- z)/y - z = -dt Integrating, = > y - z = A e^-t because y = z - A e^-t Now from (iii) z = x + y = 20 y + y = 20 (z + A e^-t) + z - A e^-1 = > z - z - 20 z = 19 e^-t solving we get z = c_1 e^5t +_2 e^-4t - 19/18 e^-t because y = c_1 e^5t + c_2 e^-4t - 19/18 e^-t + A e^-t (from circledtimes) from (iii), x = dz/dt - y = 5 c_1 e^5t - 4c_2 e^-4t + 19/18 e^-t - c_1 e^5t - c_2 e^-4t + 19/18 e^-t = 4 c_1 e^5t - 5 c_2 e^-4t + 19/9 e^-t - A e^-t because the solutions are: x(t) = 4 c_1 e^5t - 5 c_2 e^-4t + 19/9 e^-t - A e^-t y (t) = c_1 e^5t + c_2 e^-4t - 19/18 e^-t + A e^-t z (t) = c_1 e^5t + c_2 e^-t - 19/18 e^-t