Lecture Notes in Differential Equations - Bruce E. Shapiro

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Theorem 22.5. Properties of the Linear Differential Operator. Let
and denote its characteristic polynomial by
Then for any function y(t) and any scalar r,
L = aD 2 + bD + c (22.48)
P (r) = ar 2 + br + c (22.49)
Ly = P (D)y (22.50)
Le rt = P (r)e rt (22.51)
Lye rt = e rt P (D + r)y (22.52)
Proof. To demonstrate (22.50) we replace r by D in (22.49):
To derive (22.51), we calculate
P (D)y = (aD 2 + bD + c)y = Ly (22.53)
Le rt = a(e rt ) ′′ + b(e rt ) ′ + c(e rt (22.54)
= ar 2 e rt + bre rt + ce rt (22.55)
= (ar 2 + br + c)e rt (22.56)
= P (r)e rt (22.57)
To derive (22.52) we apply the differential operator to the product ye rt and
expand all of the derivatives:
Lye rt = aD 2 (ye rt ) + bD(ye rt ) + cye rt (22.58)
= a(ye rt ) ′′ + b(ye rt ) ′ + cye rt (22.59)
= a(y ′ e rt + rye rt ) ′ + b(y ′ e rt + rye rt ) + cye rt (22.60)
= a(y ′′ e rt + 2ry ′ e rt + r 2 ye rt )
+ b(y ′ e rt + rye rt ) + cye rt (22.61)
= e rt [a(y ′′ + 2ry ′ + r 2 y) + b(y ′ + ry) + cy] (22.62)
= e rt [ a(D 2 + 2Dr + r 2 )y + b(D + r)y + cy ] (22.63)
= e rt [ a(D + r) 2 y + b(D + r)y + cy ] (22.64)
= e rt [ a(D + r) 2 + b(D + r) + c ] y (22.65)
= e rt P (D + r)y (22.66)