At any time t

x=vt

dy/dt = u = x(a-x)*(v/a²)=vt(a-vt)*(v/a²)

y = (at²/2-vt³/3)*(v/a)²

Putting t = x/v we get y = (x²/2a)-(x3/3a²) (there is a misprint in your answer)

 

b) River is crossed when x= a So t= a/v

 

c) At x = a u=0

So velocity is v i

 

d) At x= a, y= a/6 So displacement = a i + a/6 j