2. With respect to a fixed origin \(O\), the position vector, \(\mathbf { r }\) metres, of a particle \(P\) at time \(t\) seconds satisfies $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + \mathbf { r } = ( \mathbf { i } - \mathbf { j } ) \mathrm { e } ^ { - 2 t } .$$ Given that \(P\) is at \(O\) when \(t = 0\), find

1. \(\mathbf { r }\) in terms of \(t\),
2. a cartesian equation of the path of \(P\).