1. One application of linear transformations is to solving differential equations. Given a polynomial f€ P2, we want to consider polynomials y E P2 satisfying the differential equation 2y" + (-x + 3) y' + 2y = f on R. (*) %3D To do so, we will consider the linear transformation T: P2 → P2 defined by T(y) = 2y" + (-–x + 3) y' + 2y for each y E P2. (a) Find the matrix of T with respect to the standard basis S = {1,x, x²} on P2. (b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation 2y" + (-x + 3) y' + 2y = 0 on R. Find a basis and the dimension for the kernel of T. (c) Is T surjective? What does this tell you about the solution of the differential equation (*)?
1. One application of linear transformations is to solving differential equations. Given a polynomial f€ P2, we want to consider polynomials y E P2 satisfying the differential equation 2y" + (-x + 3) y' + 2y = f on R. (*) %3D To do so, we will consider the linear transformation T: P2 → P2 defined by T(y) = 2y" + (-–x + 3) y' + 2y for each y E P2. (a) Find the matrix of T with respect to the standard basis S = {1,x, x²} on P2. (b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation 2y" + (-x + 3) y' + 2y = 0 on R. Find a basis and the dimension for the kernel of T. (c) Is T surjective? What does this tell you about the solution of the differential equation (*)?
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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Transcribed Image Text:1. One application of linear transformations is to solving differential equations. Given a polynomial
f€ P2, we want to consider polynomials y E P2 satisfying the differential equation
2y" + (-х + 3)у'+2у 3 f on R.
(*)
To do so, we will consider the linear transformation T : P2 → P2 defined by
T(y) = 2y" + (-x+3) y' + 2y
for each y e P2.
(a) Find the matrix of T with respect to the standard basis S = {1, x, x²} on P2.
(b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation
2y" + (—х + 3) у' + 2у —D 0 on R.
Find a basis and the dimension for the kernel of T.
(c) Is T surjective? What does this tell you about the solution of the differential equation (*)?
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