Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/1693 2022 2022 Online Islamic University, Department Of Science and Information Technology, Doha, Qatar Rama Asad Nadweh In this paper we study Bäcklund transformations (BTs) for Cosgrove’s equation F-XVIII. We use the generalization of Fokas and Ablowitz method to derive BT between F-XVIII and new fourth-order ordinary differential equations of Painlevé type. Moreover we derive auto-BT and give special solutions for F-XVIII.   2023 2023 49 55 10.54216/GJMSA.050105 https://www.americaspg.com/articleinfo/33/show/1693