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Solve the inhomogeneous second order linear equation

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Solve the inhomogeneous second order linear equation by variation of parameters. See attached file.

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Solution of inhomogeneous second order linear equation by variation of parameters.

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See attached file.

SOLUTION:
(a)
The homogeneous equation
x''(t) − 2x'(t) + x(t) = 0 (0.2)
has auxiliary equation λ2 − 2λ + 1 = 0. This equation has a repeated real root _ = 1. Thus the (linearly independent) solutions are
x1(t) = e λt (= et), x2(t) = t e λt (= tet).

(b) By choosing
xp(t) = u(t)x1(t) + v(t)x2(t) = u(t)et + v(t) tet
and imposing the additional condition
u'(t)x1(t) + v'(t)x2(t) = 0
or
u'(t)et + v'(t)tet = 0 → u'(t) + tv'(t) = 0), (0.3)
we find that
xp' = u'x1 + u x1' + ...

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