  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.
