please include all steps and details Show that the functions f1 (x)=e*, f 2(x)=xe*, and f 3(x)=x*e* are linearly independent.

Name: please include all steps and details Show that the functions f1 (x)=e*
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Description: please include all steps and details Show that the functions f1 (x)=e*, f 2(x)=xe*, and f 3(x)=x*e* are linearly independent.

Answered: Show that the functions f1 (x)=e*, f 2…

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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please include all steps and details

Show that the functions f1 (x)=e*, f 2(x)=xe*, and f 3(x)=x*e* are linearly independent.

Expert Solution

Step 1

We will use $W r o n s k i a n$ function to check these functions are linearly independent or not.

we know that

$w (f_{1}, f_{2}, f_{3}) = |\begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ f'_{1} (x) & f'_{2} (x) & f'_{3} (x) \\ f''_{1} (x) & f''_{2} (x) & f''_{3} (x) \end{matrix}| . . . . . (1)$

Now,

$f_{1} (x) = e^{x} f_{2} (x) = x e^{x} f_{3} (x) = x^{2} e^{x} f'_{1} (x) = e^{x} f'_{2} (x) = x e^{x} + e^{x} f'_{3} (x) = x^{2} e^{x} + 2 x e^{x} f''_{1} (x) = e^{x} f''_{2} (x) = x e^{x} + 2 e^{x} f''_{3} (x) = x^{2} e^{x} + 4 x e^{x} + 2 e^{x}$

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