Page 381 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 365

The functions Ui can be found by solving a series of first
order linear equations, starting from the bottom of the list.
Thus u' n = du n yields u n = c n-\e dx where c n _i is a constant.
The second last equation becomes

dx
- dun-i = c n-ie ,
u' n_ x
dx
which is first order linear with integrating factor e~ . Multi-
dx
plying the equation by this factor, we get (u n-ie~ )' = c n _i.
Hence
u n-i = (c n _ 2 + c n-ix)e dx

where c n_2 is another constant. The next equation yields

dx
- = (c n-2 + c n-ix)e ,
u' n_ 2 du n- 2
dx
which is also first order linear with integrating factor e~ . It
can be solved to give

/ . c n - 2 , c n - l 2\ dx
U n-2 = (c n -3 + ~jr X + ~^T X ) e '

where c n„2 is constant. Continuing in this manner, we find
that the function « n _i is given by

^•n—i — (Cn — i — 1 i r~| X "T ' " ' ~r r. X )C ,
1! i\
where the Cj are constants. The original functions j/j can then
be calculated by using the equation Y — SU.

Example 9.4.10
Solve the linear system of differential equations below using
Jordan normal form:

y[ = 3yi + y 2
= -yi + V2
y 2
=
y' 3 2y 3