Testing a Set of Solutions for Linear Independence Determine whether {1, cos [latex]x,[/latex] sin [latex]x[/latex]} is a set of linearly independent solutions of the linear homogeneous differential equation [latex]y^{\prime\prime\prime } + y^{\prime} =0[/latex].

Begin by observing that each of the functions is a solution of [latex]y^{\prime\prime\prime } + y^{\prime} =0[/latex]. (Try checking this.) Next, testing for linear independence produces the Wronskian of the three functions, as follows. [latex]W = \left | \begin{matrix} 1 & cos x & sin x \\ 0 & -sinx & cos x \\ 0 & -cos x & -sinx \end{matrix} \right | [/latex] = sin² [latex]x[/latex] + cos² [latex]x[/latex] = 1 Because [latex]W[/latex] is not identically equal to zero, the set {[latex]1, cos x, sin x[/latex]} is linearly independent. Moreover, because this set consists of three linearly independent solutions of a third-order linear homogeneous differential equation, the general solution is y = [latex]C_1 + C_2[/latex] cos [latex]x + C_3[/latex] sin [latex]x[/latex] where [latex]C_1, C_2,[/latex] and are real numbers.

Question 4.8.3: Testing a Set of Solutions for Linear Independence Determine...

Elementary Linear Algebra

Testing a Set of Solutions

for Linear Independence

Determine whether {1, cos $x,$ sin $x$ } is a set of linearly independent solutions of the linear homogeneous differential equation

$y^{\prime\prime\prime } + y^{\prime} =0$ .

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Begin by observing that each of the functions is a solution of $y^{\prime\prime\prime } + y^{\prime} =0$ . (Try checking this.) Next, testing for linear independence produces the Wronskian of the three functions, as follows.

W = \left | \begin{matrix} 1 & cos x & sin x \\ 0 & -sinx & cos x \\ 0 & -cos x & -sinx \end{matrix} \right |

= sin² $x$ + cos² $x$ = 1

Because $W$ is not identically equal to zero, the set

{ $1, cos x, sin x$ }
is linearly independent. Moreover, because this set consists of three linearly independent solutions of a third-order linear homogeneous differential equation, the general solution is

y = $C_1 + C_2$ cos $x + C_3$ sin $x$
where $C_1, C_2,$ and are real numbers.