If $\displaystyle y_1 $ and $\displaystyle y_2$ satisfy

$\displaystyle y_1 '' + \lambda_1 y_1 = 0 \ \ \ y_1 ' (0) = 0, \ \ y_1(L) = 0$

$\displaystyle y_2 '' + \lambda_2 y_2 = 0 \ \ \ y_2 ' (0) = 0, \ \ y_2(L) = 0$

Without determining the explicit form of both $\displaystyle y_1 $ and $\displaystyle y_2$, show that

$\displaystyle \int_0^L y_1 y_2 \ dx = 0 $ if $\displaystyle \lambda_1 \neq \lambda_2$ .