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8.3: Applications to Systems of Linear Differential Equations 295

is not diagonalizable, but it can be triangularized. In fact it
was shown in Example 8.1.6 that

T = S->AS=(l \

where S = I J. Put U = S Y and write ui, u 2 for

the entries of U. Then Y = SU and Y' = SU'. The equation
Y' = AY now becomes U' = TU. This yields the triangular
system
u[ = 2ui + u<i

u' 2 = 2u2
Solving the second equation, we find that u 2 = c 2 e 2x with
C2 an arbitrary constant. Now substitute for u^ in the first
equation to get
2x
u[ - 2ui = c 2e .
This is a first order linear equation which can be solved by a
standard method: multiply both sides of the equation by the
"integrating factor"

f -2dx -2x

2x
The equation then becomes (uie~ )' = c 2, whence u\e~ 2x =
c 2x + ci, with c\ another arbitrary constant. Thus u\ —
2x
c 2xe 2x + cxe . To find the original functions j/i and y 2, we
form the product

2x C l C 2 X
Y = SU = e ( c + ^
\ i +c 2{x + 1)
Thus the general solution of the system is

2x
J/i = (ci +c 2x)e ,
2x
y2 = (ci + c 2(x + l))e