Determine if the following sets of functions are linearly independent using theWronskian.(a)(b)(c)f(x) = sin(x), g(x) = cos(r),f(x)=e³, g(x)=e=³(x+1),f(x)=x², g(x)=√x.

(a) Recall the Wronskian of two function is Now, are linearly independent.(b) Let Then,…

Answered: Determine if the following sets of…

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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Question

Determine if the following sets of functions are linearly independent using the
Wronskian.
(a)
(b)
(c)
f(x) = sin(x), g(x) = cos(r),
f(x)=e³, g(x)=e=³(x+1),
f(x)=x², g(x)=√x.

Expert Solution

Step 1: ''Introduction to the solution''

(a) $L e t space space space f left parenthesis x right parenthesis equals sin left parenthesis x right parenthesis comma space g left parenthesis x right parenthesis equals cos left parenthesis x right parenthesis comma space x element of straight real numbers$

Recall the Wronskian of two function $space f space a n d space g space$ is $W left parenthesis f comma g right parenthesis equals open vertical bar table row cell f left parenthesis x right parenthesis end cell cell g left parenthesis x right parenthesis end cell row cell f apostrophe left parenthesis x right parenthesis end cell cell g apostrophe left parenthesis x right parenthesis end cell end table close vertical bar$

Now, $W left parenthesis f comma g right parenthesis equals open vertical bar table row cell sin left parenthesis x right parenthesis end cell cell cos left parenthesis x right parenthesis end cell row cell cos left parenthesis x right parenthesis end cell cell negative sin left parenthesis x right parenthesis end cell end table close vertical bar equals negative open parentheses sin squared x plus cos squared x close parentheses equals negative 1 not equal to 0$

$rightwards double arrow$ $f space a n d space space g space space$ are linearly independent.

(b) Let $f left parenthesis x right parenthesis equals e to the power of negative 3 x end exponent comma space g left parenthesis x right parenthesis equals e to the power of negative 3 open parentheses x plus 1 close parentheses end exponent$

Then, Wronskian of $f$ and $g$ is $W open parentheses f comma g close parentheses equals open vertical bar table row cell f left parenthesis x right parenthesis end cell cell g left parenthesis x right parenthesis end cell row cell f apostrophe left parenthesis x right parenthesis end cell cell g apostrophe left parenthesis x right parenthesis end cell end table close vertical bar equals open vertical bar table row cell e to the power of negative 3 x end exponent end cell cell e to the power of negative 3 left parenthesis x plus 1 right parenthesis end exponent end cell row cell negative 3 e to the power of negative 3 x end exponent end cell cell negative 3 e to the power of negative 3 left parenthesis x plus 1 right parenthesis end exponent end cell end table close vertical bar equals open parentheses negative 3 space e to the power of negative 3 x end exponent space e to the power of negative 3 left parenthesis x plus 1 right parenthesis end exponent plus 3 space e to the power of negative 3 x end exponent space e to the power of negative 3 left parenthesis x plus 1 right parenthesis end exponent close parentheses equals 0 rightwards double arrow f space a n d space g space space space a r e space space l i n e a r l y space space d e p e n d e n t.$