An equation of the curve for which the 

portion of y-axis cut off between the origin and the tangent 

varies as the cube of the abscissa of the point of contact is

The portion of $y$-axis cut off between the

origin and the tangent is $y-x \frac{\mathrm{d}y}{\mathrm{d}x}$ . According to the given

condition $y-x \frac{\mathrm{d}y}{\mathrm{d}x}=K{x}^3$ (K is constant of proportionality)

$\Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x}-\frac{1}{x} y=-K{x}^2$ . This is a linear equation whose

                                       I.F. is 1/x.

Hence $\frac{\mathrm{d}}{\mathrm{d}x}(y/x)=-Kx\Rightarrow y=-K{x}^3/2+Cx$.