5 The substitution \(y = u ^ { k }\), where \(k\) is an integer, is to be used to solve the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } y ^ { 2 }$$ by changing it into an equation (B) in the variables \(u\) and \(x\).

1. Show that equation (B) may be written in the form $$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 3 } { k x } u = \frac { 1 } { k } x u ^ { k + 1 }$$
2. Write down the value of \(k\) for which the integrating factor method may be used to solve equation (B).
3. Using this value of \(k\), solve equation (B) and hence find the general solution of equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).
   (a) The set of polynomials \(\{ a x + b \}\), where \(a , b \in \mathbb { R }\), is denoted by \(P\). Assuming that the associativity property holds, prove that \(P\), under addition, is a group.
   (b) The set of polynomials \(\{ a x + b \}\), where \(a , b \in \{ 0,1,2 \}\), is denoted by \(Q\). It is given that \(Q\), under addition modulo 3 , is a group, denoted by \(( Q , + ( \bmod 3 ) )\).
4. State the order of the group.
5. Write down the inverse of the element \(2 x + 1\).
6. \(\mathrm { q } ( x ) = a x + b\) is any element of \(Q\) other than the identity. Find the order of \(\mathrm { q } ( x )\) and hence determine whether \(( Q , + ( \bmod 3 ) )\) is a cyclic group.