Fundamental Solution Set

We have learnt that Superposition Principle allows us to find new solutions from linear combinations of known solutions. However, we may be interested in fundamental solution set which is a set of solutions that can be used as the building blocks to find all other solutions.

A fundamental set of solutions is a set of linearly indepenent solutions to a DE on an interval, \(I\). A linearly independent set of functions on interval \(I\) only has a trivial solution to the following equation:

\( c_1f_1(x) + c_2f_2(x) + \dots + c_nf_n(x) = 0\)

Are the functions \( x^2 \) and \(2x^2 \) linearly independent?


One method to determine if functions are linearly independent is to check the Wronskian. The Wronskian of a group of functions is defined as:

\( W(f_1, f_2, \dots f_n) = \begin{vmatrix} y_1 & y_2 & y_3 \\ y^{'}_1 & y^{'}_2 & y^{'}_3 \\ y^{''}_1 & y^{''}_2 & y^{''}_3 \end{vmatrix} \)

A set of functions is linearly independent on interval \(I\) iif:

\( W(f_1, f_2, \dots f_n) \ne 0 \)

Confirm the set of solutions \( \{e^{-10x}, e^{5x}\} \) is a fundamental solution set to the DE \(y^{''} + 5y^{'} - 50 y = 0 \)

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