Understanding the Wronskian Determinant

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SUMMARY

The Wronskian determinant is defined for two functions \( f_1 \) and \( f_2 \) as \( W(f_1, f_2) = \begin{vmatrix} f_1 & f_2 \\ f_1' & f_2' \end{vmatrix} \). If the Wronskian is non-zero over an interval, it indicates that the functions \( f_1 \) and \( f_2 \) are linearly independent on that interval. This determinant plays a crucial role in the study of differential equations and function independence.

PREREQUISITES
  • Understanding of determinants in linear algebra
  • Knowledge of calculus, specifically derivatives
  • Familiarity with linear independence concepts
  • Basic understanding of differential equations
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  • Study the properties of determinants in linear algebra
  • Learn about linear independence in the context of function spaces
  • Explore applications of the Wronskian in solving differential equations
  • Investigate examples of Wronskian calculations for specific functions
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Students of mathematics, particularly those studying calculus and differential equations, as well as educators looking to explain the concept of linear independence through the Wronskian determinant.

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Hi guys! How's it going? OK so I'm like totally stuck and I have no idea how to do this. View attachment 4011
 

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The Wronskian of two functions $ f_1$ and $ f_2$ is defined as
\[
W (f_1, f_2 ) =
\left | {\begin{array}{cc}
f_1 & f_2 \\
f_{1}^{'} & f_{2}^{'} \\
\end{array} } \right |
\]
Using this definition, what then is the Wronskian determinant?

Then, if the determinant is never 0 on the interval, the functions are linearly independent.
 
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