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The set: V = {Ae² sin (3x) + Be2 cos (3x): A, B = R} consists of all the solutions of the differential equation: d²y dx² dy 4- + 13y = 0 dx and forms a vector space over R with the usual addition and scalar multiplication operations.

The set: V = {Ae² sin (3x) + Be2 cos (3x): A, B = R} consists of all the solutions of the differential equation: d²y dx² dy 4- + 13y = 0 dx and forms a vector space over R with the usual addition and scalar multiplication operations.

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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(i) Let f: V→ V be the function that takes a solution of the form Ae²* sin(3x) + Be2* cos(3x) to the solution obtained by substituting x+ for x. Show that f is a linear map. (ii) Find a matrix that represents f relative to the basis S = {e² sin (3x), e²* cos(3x)}. (iii) Show that U = {Ae²* sin(3x): A = R} is a subspace of V.
Transcribed Image Text:(i) Let f: V→ V be the function that takes a solution of the form Ae²* sin(3x) + Be2* cos(3x) to the solution obtained by substituting x+ for x. Show that f is a linear map. (ii) Find a matrix that represents f relative to the basis S = {e² sin (3x), e²* cos(3x)}. (iii) Show that U = {Ae²* sin(3x): A = R} is a subspace of V.
The set: V = {Ae² sin (3x) + Be² cos (3x): A, B = R} consists of all the solutions of the differential equation: d'y dx² dy 4 + 13y = 0 dx and forms a vector space over R with the usual addition and scalar multiplication operations.
Transcribed Image Text:The set: V = {Ae² sin (3x) + Be² cos (3x): A, B = R} consists of all the solutions of the differential equation: d'y dx² dy 4 + 13y = 0 dx and forms a vector space over R with the usual addition and scalar multiplication operations.
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