7 It is given that \(y = x ^ { 2 } w\) and
$$x ^ { 2 } \frac { d ^ { 2 } w } { d x ^ { 2 } } + 4 x ( x + 1 ) \frac { d w } { d x } + \left( 5 x ^ { 2 } + 8 x + 2 \right) w = 5 x ^ { 2 } + 4 x + 2$$
- Show that
$$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 4 \frac { d y } { d x } + 5 y = 5 x ^ { 2 } + 4 x + 2$$
- Find the general solution for \(w\) in terms of \(x\).