Infinite families of solutions for $A^3+B^3=C^3+D^3$ and $A^4+B^4+C^4+D^4+E^4=F^4$

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation $A^3+B^3=C^3+D^3$. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation $A^4+B^4+C^4+D^4+E^4=F^4$ in the spirit of Ramanujan.