V Let be the set of all functions of the form f(x) = ae* +bVx , a,b € R This set is a subset of the set of all continuous functions on (0,0). it is also a subspace. (a) Verify the closure axioms explicitly for functions in this form. (b) The set is an obvious spanning set, since the given function form is linear combinations of that set. You need to show independence. While use of the Wronskian is a result from Differential Equations, it is NOT a result that is part of the body of theorems for this fe", V class, and so you can't bring it in here. Verify the independence of by directly setting up the equation that defines independence. That should give you an equation with Ci, C2 and the function of *. The way to test this is to just pick two arbitrary values of X and use them to generate an easy system of two linear equations that can be solved for C and C2.
V Let be the set of all functions of the form f(x) = ae* +bVx , a,b € R This set is a subset of the set of all continuous functions on (0,0). it is also a subspace. (a) Verify the closure axioms explicitly for functions in this form. (b) The set is an obvious spanning set, since the given function form is linear combinations of that set. You need to show independence. While use of the Wronskian is a result from Differential Equations, it is NOT a result that is part of the body of theorems for this fe", V class, and so you can't bring it in here. Verify the independence of by directly setting up the equation that defines independence. That should give you an equation with Ci, C2 and the function of *. The way to test this is to just pick two arbitrary values of X and use them to generate an easy system of two linear equations that can be solved for C and C2.
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:V
Let
be the set of all functions of the form f(x) = ae* +b\x , a,D ER This set is a subset of the set
of all continuous functions on (0,"). it is also a subspace.
(a) Verify the closure axioms explicitly for functions in this form.
(b) The set S is an obvious spanning set, since the given function form is linear
combinations of that set. You need to show independence. While use of the Wronskian is a
result from Differential Equations, it is NOT a result that is part of the body of theorems for this
class, and so you can't bring it in here. Verify the independence of *S by directly setting
up the equation that defines independence. That should give you an equation with C1, C2 and
the function of X. The way to test this is to just pick two arbitrary values of X and use them to
generate an easy system of two linear equations that can be solved for C and C2.
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