Answer
FALSE
Work Step by Step
By linearity of differentiation, $(c_{1}y_{1}+c_{2}y_{2})''=c_{1}y_{2}''+c_{2}y_{2}''$ and $(c_{1}y_{1}+c_{2}y_{2})'=c_{1}y_{1}'+c_{2}y_{2}'$,so
$(c_{1}y_{1}+c_{2}y_{2})''+6(c_{1}y_{1}+c_{2}y_{2})'+5(c_{1}y_{1}+c_{2}y_{2})$
$=c_{1}(y_{1}''+6y_{1}'+5y_{1})+c_{2}(y_{2}''+6y_{2}'+5y_{2})$
$=x+x=2x$
Thus $c_{1}y_{1}+c_{2}y_{2}$ solves $y''+6y'+5y=2x\ne x$
