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.