Page 310 - A Course in Linear Algebra with Applications

2 9 4 Chapter Eight: Eigenvectors and Eigenvalues

be solutions of the system Y' = AY. Thus we obtain two real
solutions from the single complex solution Y, by taking the
real and imaginary parts of Y; these are respectively

/—e^sin x\ , , . /e^cos x
\ e cos x J \ e x sin x
Now Y\ and Y 2 are easily seen to be linearly independent solu-
tions; therefore the general solution of the system is obtained
by taking an arbitrary linear combination of these:

Cl S i n x + C2 C S X
Y = c 1Y 1 + c 2Y 2 = e*( ~ °
y c\ cos x + c 2 sin x
where c\ and c 2 are arbitrary real constants. Hence
x
J/i =e (—ci sin x + c 2 cos x)
x
= e (c\ cos a: + C2 sin a;)
y 2
Of course the success of the method employed in the last
two examples depended entirely upon the fact that A is diag-
onalizable. However, should this not be the case, one can still
treat the system of differential equations by triangularizing
the coefficient matrix and solving the resulting triangular sys-
tem using back substitution, rather as was done for systems
of linear recurrences in 8.2.

Example 8.3.3
Solve the linear system of differential equations

2/i = 3 / i + 2/2
3
2/2 = -2/1 + 2/2
In this case the coefficient matrix