- (a) Show that the substitution \(y = v x\) transforms the differential equation
$$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 3 } + y ^ { 3 }$$
into the differential equation
$$3 v ^ { 2 } x \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1 - 2 v ^ { 3 }$$
(b) By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = \mathrm { f } ( x )\).
Given that \(y = 2\) at \(x = 1\),
(c) find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 1\)