10 A particle P , of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac { k m g } { x ^ { 2 } }\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\).
P is initially projected in a direction directly away from O . The velocity of P is first zero at a point A which is a distance \(a\) from O .

1. Show that the velocity \(v\) of P , when P is moving away from O , satisfies the differential equation $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k g } { x ^ { 2 } } + 2 \mu g = 0$$
2. Verify that $$v ^ { 2 } = 2 g k \left( \frac { 1 } { x } - \frac { 1 } { a } \right) + 2 \mu g ( a - x )$$
3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A .