The solution of y=x(dydx)+(dydx)3 are given by (where p=dy/dx and k is constant).
Put p=dydx in the given equation y=x(dydx)+(dydx)3.
y=px+p3
dydx=dpdx(x)+p+3p2dpdx
p=xdpdx+p+3p2dpdx
dpdx(x+3p2)=0
dpdx=0;x+3p2=0
p=0;p2=−x3
dydx=0
y=0