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Dg = 2g + h Dh = g+h a) Find the differential equation D² f + aDƒ+ßf = 0 that functions in V must satisfy. b) Find the matrix A of D in this basis. c) Verify that for the scalars a and B that you found: A² + aA+ BI = 0.

Dg = 2g + h Dh = g+h a) Find the differential equation D² f + aDƒ+ßf = 0 that functions in V must satisfy. b) Find the matrix A of D in this basis. c) Verify that for the scalars a and B that you found: A² + aA+ BI = 0.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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Please help. This problem involves bases and linear transformations. Thank you.

Problem 10. Suppose that V is the vector space of solutions of a second order differential equation with constant coefficients D² f+aDf+Bƒ = 0. Suppose that we are given a basis B = {g(x), h(x)} of the space of solutions V, consisting
Transcribed Image Text:Problem 10. Suppose that V is the vector space of solutions of a second order differential equation with constant coefficients D² f+aDf+Bƒ = 0. Suppose that we are given a basis B = {g(x), h(x)} of the space of solutions V, consisting
of functions with derivatives of all orders in R. Assume that the linear transformation D = d.x * satisfies: Dg = 2g + h Dh = g+h a) Find the differential equation D² f + aDƒ + ßf = 0 that functions in V must satisfy. b) Find the matrix A of D in this basis. c) Verify that for the scalars a and B that you found: A² + aA+ BI = 0. Hint: Both g, h need to satisfy the differential equation. Use the informa- tion about Dg and Dh which is written in terms of g, h to try to find the differential equation that must be satisfied.
Transcribed Image Text:of functions with derivatives of all orders in R. Assume that the linear transformation D = d.x * satisfies: Dg = 2g + h Dh = g+h a) Find the differential equation D² f + aDƒ + ßf = 0 that functions in V must satisfy. b) Find the matrix A of D in this basis. c) Verify that for the scalars a and B that you found: A² + aA+ BI = 0. Hint: Both g, h need to satisfy the differential equation. Use the informa- tion about Dg and Dh which is written in terms of g, h to try to find the differential equation that must be satisfied.
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