Find the general solution of the system...?

Question

dx/dt = 2x - 7y
dy/dt = 5x + 10y + 4z
dz/dt = 5y + 2z

Thanks very much! :)

Answer 1

Let D denote the differential operator, and rewrite the system as
(1) (D - 2)x + 7y = 0,
(2) -5x + (D - 10)y - 4z = 0,
(3) -5y + (D - 2)z = 0.
Now, operate on (2) with (D - 2) and on (3) with 4 and add the results to get
(4) (-5D + 10)x + (D^2 - 12)y = 0.
Operate now on (1) with -5 and find
(5) (5D - 10)x - 35y = 0.
By adding (4) and (5) we find (D^2 - 12D - 35)y = 0. That is a standard homogeneous differential equation, which you may solve for y. Then use that value of y in (1) to obtain a typical nonhomogeneous equation, which you should solve for x. Finally, use the known value for y in (3) and solve for z.