Question

Work out the Wronskian for the following pairs of functions and state whether the are linearly dependent or linearly independent in the interval x €(01). (Please Show All Work!!! Y; (x)=sin(x); y2(x)= xsin(x) Please Show All Your Work, Or Risk Losing All Points x+(0, ') Votre Pom

Answer 1

The Wronskian determinant is

$\begin{vmatrix}\sin x&x\sin x\\\cos x&x\cos x+\sin x\end{vmatrix}=\sin x(x\cos x+\sin x) - x\sin x\cos x=\sin^2x$

which is non-zero for all $x\in(0,1)$, so the solutions are linearly independent.