To get solutions of x^2=y^3 one can look for solutions of y=x^(2/3). Taking x=t^3 for some other integer t one can guarantee that y=t^(3*2/3)=t^2 is an integer. So my solutions would be
x=0,y=0
x=1,y=1
x=-1,y=1
x=8,y=4
x=-8,y=4
...
To get solutions of x^2=y^3 one can look for solutions of y=x^(2/3). Taking x=t^3 for some other integer t one can guarantee that y=t^(3*2/3)=t^2 is an integer. So my solutions would be
x=0,y=0
x=1,y=1
x=-1,y=1
x=8,y=4
x=-8,y=4
...