Solve using Wronskian Determine whether the functions f1(x) = e2", f2(x) = e-2", and f3(x) = sinh 2x are linearlyindependent on the interval (-o, 0).

Answered: Determine whether the functions f1(r) =…

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

Solve using Wronskian

Determine whether the functions f1(x) = e2", f2(x) = e-2", and f3(x) = sinh 2x are linearly
independent on the interval (-o, 0).

Expert Solution

Step 1

To determine whether the functions $f_{1} (x) = e^{2 x}, f_{2} (x) = e^{- 2 x}, and f_{3} (x) = sinh 2 x$ are linearly independent on the interval $(- \infty, \infty)$ .

Pre-requisite : $f_{1}, f_{2}, and f_{3}$ are linearly dependent if and only if the Wronskian is identically zero for all $x$ , that is,

$W (f_{1}, f_{2}, f_{3}) = 0 \forall x$

where $W (f_{1}, f_{2}, f_{3}) = |\begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ f_{1}^{'} (x) & f_{2}^{'} (x) & f_{3}^{'} (x) \\ f_{1}^{''} (x) & f_{2}^{''} (x) & f_{3}^{''} (x) \end{matrix}|$