Exercises on differential equations and eAt
Problem 23.1: (6.3 #14.a Introduction to Linear Algebra: Strang) The matrix in this question is skew-symmetric (AT = −A) :
du = ⎡ −0c 0c −ba ⎦⎤ u or uu2 = au3 − cu13
⎣ dt b −a 0 u .
Find the derivative of ||u(t)||2 using the definition:
||u(t)||2 = u12 + u22 + u32.
What does this tell you about the rate of change of the length of u? What does this tell you about the range of values of u(t)?
Problem 23.2: (6.3 #24.) Write A as SΛS−1. Multiply SeΛtS−1
to find the matrix exponential eAt. Check your work by evaluating eAt and the derivative of eAt when t = 0.
