Update
george2079 pointed out that using approximate numbers created a problem with the solution for high values of t
. Replacing the approximate value for gamma
with a rational number, the problem is stable at high values, including infinity.
γ = 267500000;
t1 = 1/10;
t2 = 1/100;
b0 = 2;
m0 = 1/10;
b1 = 1/10000000;
DSolve
works with your equations.
The first differential equation yields a constant value for mx[t]
of zero so I will skip it.
The second differential equation gives:
solYZ =
DSolve[{my'[t] == γ*mz[t]*b1 - my[t]/t2,
mz'[t] == -γ*my[t]*b1 + (m0 - mz[t])/t1, my[0] == 0,
mz[0] == m0}, {my[t], mz[t]}, t]
(* {{my[t] -> (1/11501679980)
107 E^(-55 t - (Sqrt[20951] t)/4 -
1/4 (-220 + Sqrt[20951]) t) (1676080 E^((Sqrt[20951] t)/2) -
838040 E^(1/4 (-220 + Sqrt[20951]) t) -
18649 Sqrt[20951] E^(1/4 (-220 + Sqrt[20951]) t) -
838040 E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t) +
18649 Sqrt[20951]
E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t)),
mz[t] -> (1/11501679980)
E^(-55 t - (Sqrt[20951] t)/4 -
1/4 (-220 + Sqrt[20951]) t) (670432000 E^((Sqrt[20951] t)/2) +
239867999 E^(1/4 (-220 + Sqrt[20951]) t) -
2518780 Sqrt[20951] E^(1/4 (-220 + Sqrt[20951]) t) +
239867999 E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t) +
2518780 Sqrt[20951]
E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t))}} *)
We can create functions out of the solutions in a number of ways. Below is one method:
my[t_] = solYZ[[1, 1, 2]]
(* (1/11501679980)107 E^(-55 t - (Sqrt[20951] t)/4 -
1/4 (-220 + Sqrt[20951]) t) (1676080 E^((Sqrt[20951] t)/2) -
838040 E^(1/4 (-220 + Sqrt[20951]) t) -
18649 Sqrt[20951] E^(1/4 (-220 + Sqrt[20951]) t) -
838040 E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t) +
18649 Sqrt[20951]
E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t)) *)
and for mz[t]
mz[t_] = solYZ[[1, 2, 2]]
(* (1/11501679980)E^(-55 t - (Sqrt[20951] t)/4 -
1/4 (-220 + Sqrt[20951]) t) (670432000 E^((Sqrt[20951] t)/2) +
239867999 E^(1/4 (-220 + Sqrt[20951]) t) -
2518780 Sqrt[20951] E^(1/4 (-220 + Sqrt[20951]) t) +
239867999 E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t) +
2518780 Sqrt[20951]
E^((Sqrt[20951] t)/2 + 1/4 (-220 + Sqrt[20951]) t)) *)
Now we can plot it:
Plot[{my[t], mz[t]}, {t, 0, 1},
PlotStyle -> {{Thick, Blue}, {Thick, Red}}]

It appears to quickly approach a constant. If you take the limits you will find:
Limit[my[t], t -> ∞]
(* 428/27449 *)
Limit[mz[t], t -> ∞]
(* 1600/27449 *)
%
incorrectly. (first usage tries to getMX
from theMy,Mz
solution for example) I'd suggest actually assiging theNDSolve
results to symbols and using those instead of relying on output history (%
) $\endgroup$