Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations \(\begin{aligned} x_{n+1}=\frac{x_{n-2}x_{n-3}}{ax_{n}+bx_{n-3}}, \ n\ge 0, \end{aligned}\) and \(\begin{aligned} x_{n+1}=\frac{x_{n-2}x_{n-3}}{-ax_{n}+bx_{n-3}}, \ n\ge 0, \end{aligned}\) with positive arbitrary real parameters a, b and arbitrary real initial conditions, as well as study the qualitative behaviors for each. For the first equation, we show that every admissible solution converges to a period-3 solution when \(a+b=1\) . For the second equation, we show that every admissible solution converges to zero if \(b>2\) when \(b^2\ge 4a\) . When \(b^2<4a\) , we show the existence of periodic solutions under certain conditions. We introduce the forbidden sets as well as provide some illustrative examples for the above-mentioned equations.
