Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 4 - Higher Order Linear Equations - 4.1 General Theory of nth Order Linear Equations - Problems - Page 224: 9


$f_{1},f_{2},f_{3},f_{4}$ are linearly dependent.

Work Step by Step

$af_{1}(t)+bf_{2}(t)+cf_{3}(t)+df_{4}=\;a(2t-3)\;+b(t^2+1)\;+c(2t^2-t)\;+d(t^2+t+1)=\;0\\\\$ $(-3a+b+d)+\;(2a-c+d)t+(b+2c+d)t^2=0\\\\$ By the polynomial equality theorem: $\left\{\begin{matrix} -3a+b+d=0 \\ 2a-c+d=0 \\ b+2c+d=0 \end{matrix}\right. \rightarrow \;\;\;\;a=\frac{-2}{7}\;\;b=\frac{-13}{7}\;\;c=\frac{3}{7}\;\;c=\frac{3}{7}\;\;d=1\\\\$ so, $f_{1},f_{2},f_{3},f_{4}$ are linearly dependent.
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