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r(a0, a1, a2, . . ) =) n! n20 (a) Let k E R. Show that y((1, k, k² , k³, . )) is the expontial func- tion ekr. (b) Find all solutions to the recurrence relation a, = ka, 1. (c) Find all solutions to the differential equation y'(t) = ky(t). (d) How are these two sets of solutions related by y?

r(a0, a1, a2, . . ) =) n! n20 (a) Let k E R. Show that y((1, k, k² , k³, . )) is the expontial func- tion ekr. (b) Find all solutions to the recurrence relation a, = ka, 1. (c) Find all solutions to the differential equation y'(t) = ky(t). (d) How are these two sets of solutions related by y?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 7.6 Let S = {(ao, a1, a2, . . )} be the set of all sequences of real numbers, and let P = {Eo Cnr"} be the set of all power series in r with real coefficients. Define the function y : S → P by %3D y(ao, a1, a2, . . .) = 1". n! n20 (a) Let k E R. Show that y((1, k, k², k³, . )) is the expontial func- tion ekz. (b) Find all solutions to the recurrence relation a, = ka, 1- (c) Find all solutions to the differential equation y'(t) = ky(t). (d) How are these two sets of solutions related by y ?
Transcribed Image Text:Problem 7.6 Let S = {(ao, a1, a2, . . )} be the set of all sequences of real numbers, and let P = {Eo Cnr"} be the set of all power series in r with real coefficients. Define the function y : S → P by %3D y(ao, a1, a2, . . .) = 1". n! n20 (a) Let k E R. Show that y((1, k, k², k³, . )) is the expontial func- tion ekz. (b) Find all solutions to the recurrence relation a, = ka, 1- (c) Find all solutions to the differential equation y'(t) = ky(t). (d) How are these two sets of solutions related by y ?
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