Lecture Notes in Differential Equations - Bruce E. Shapiro

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L n+1 (e rt y) = e rt a n (D + r − r 1 )(D + r − r 2 ) · · · (D + r − r n+1 )y
= e rt P n+1 (D + r)y
(27.17)
which proves the assertion for n + 1, completing the inductive proof. .
The general solution to (27.1) is
where
y = y H + y P (27.18)
y H = C 1 y H,1 + C 2 y H,2 + · · · + C n y H,n (27.19)
and the y H,i are linearly independent solutions of the homogeneous equation
L n y = 0. If L is n-th order then there will be n linearly independent
solutions; taken together, any set of n linearly independent solutions are
called a fundamental set of solutions. Note that the set is not unique,
because if y is a solution the differential equation then so is cy for any
constant c.
Superposition and Subtraction
As before with second order equations, we have a principle of superposition
and a subtraction principle.
Theorem 27.1. (Principle of Superposition.) If u(t) and v(t) are any two
solutions of L n y = 0 then any linear combination w(t) = c 1 u(t) + c 2 v(t) is
also a solution of L n y = 0.
Theorem 27.2. (Subtraction Principle)If u(t) and v(t) are any solutions
to L n y = f(t) then w(t) = u(t) − v(t) is a solution to the homogeneous
equation L n y = 0.