Express \(\frac { 1 } { ( 1 + x ) ( 2 + x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 2 + x }\), where \(A\) and \(B\) are integers.
Use the substitution \(u = \frac { \mathrm { d } y } { \mathrm {~d} x }\) to solve the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { 1 } { ( 1 + x ) ( 2 + x ) } \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + x } { 1 + x }$$
given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\). [0pt]
[11 marks]