Question

Use the properties of a Cauchy-Euler system to find a general solution of the given system. 3 7 tx'(t)= X(t), t> - 3 13 For t>0, any Cauchy-Euler system of the form tx' = Ax with A an nxn constant matrix has nontrivial solutions of the form x(t)= t'u if and only ifr is an eigenvalue of A and u is a corresponding eigenvector. x(t) = 0

Answer 1

oll ta' (t) = 3 7 -3 13 alt), to eigen value of A 3-7 7 - 3 13-7 (3-7) (137) +21=0 72-162 +60=0 (2-10) (7-6)=0 2=10,6 eigenvector corresponding n=10 bev [33][M]-[0] - 3 [ ML] let V2=1, VFI V=1] eigen vector corresponding D= 6 be m 127-13 [37] [ ] - [] -37 - 3 7 ~ let m₂=1, m, = 7/3 m = [7]

te to 000 3 ol 7/3 X(t) = ct