In: Advanced Math
Let A =
0 | 1 |
1 | 0 |
(a) Calculate the matrix exponential e^(At). (Hint: It might
help to write down the power series expansions for the hyperbolic
functions
cosh(t) =(e^t + e^(−t))/2
and sinh(t) =(e^t −e^(−t))/2
and then try to write eAt in terms of these two functions.)
(b) Use your matrix from part (a) to solve the nonhomogeneous
initial value problem
x' =
0 | 1 |
1 | 0 |
x +
2 |
-1 |
, x(0) =
1 |
2 |
. (Hint: You might need the identity cosh^2(t)−sinh^2(t) =
1.)