Q. Suppose $u,v\&w$ are twice differentiable functions of $x$ , which satisfies the relations $au+bv+cw=0$ (where $a,b\&c$ are constants, not all zero simultaneously), the value of the determinant $\begin{vmatrix} u & v & w \\ u^{'} & v^{'} & w^{'} \\ u^{''} & v^{''} & w^{''} \end{vmatrix}$ is

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Solution:

Given $ \, au+bv+cw=0 \, \, \ldots \left(\right.1\left.\right)$
$au^{′}+bv^{′}+cw^{′}=0 \, \, \ldots \left(\right.2\left.\right)$
And $au^{′ ′}+bv^{′ ′}+cw^{′ ′}=0 \, \, \, \ldots \left(\right.3\left.\right)$
For non trivial solution (non zero) solution of a, b and c. We must have $\begin{vmatrix} u & v & w \\ u′ & v′ & w′ \\ u′′ & v′′ & w′′ \end{vmatrix}=0$

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