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The Brück conjecture states that if a nonconstant entire function 
$f$ with hyper-order 
${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value 
$a$ (counting multiplicities) with its derivative 
$f^{\prime }$, then 
$f^{\prime }-a=c(f-a)$, where 
$c$ is a nonzero constant. The conjecture has been established for entire functions with order 
${\it\sigma}(f)<+\infty$ and hyper-order 
${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case 
${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations 
$f^{(k)}+A(z)f=Q(z)$. The shared value 
$a$ is extended to be a ‘small’ function with respect to the entire function 
$f$.
