Augment the matrix with the 3 by 3 identity matrix.
You can use both
rref to gain insight
on this problem.
Set up the coefficient matrix for the general system of equations and solve it.
x is a ratio of two determinants
If you let x(t) = T(t) - T_m, then dx/dt = dT/dt.
The final principle after half a year of compounding continuously at the first rate is the initial principal for the growth which occurs compounding continuously at the second rate for the second half of the year.
Solve the exponential growth equation with the initial condition x(0) = 250. Then find t for which x(t) = 400.
The derivative at point (t,x) is x+t.
Compare numerical results with analytical results.
Notice from the line field the wide variety of possible curve shapes which can appear.
Bret Larget, email@example.com