Ordinary Differential Equations: Symmetry Solutions

In this chapter, constructing solutions using the symmetry generators will be discussed separately for first, second and higher order ordinary differential equations. Symmetries can be employed in a number of ways: Determining integrating factors (first order equations), group invariant solutions (discussed in Chapter 2 and will be discussed more systematically here), reduction of order via canonical coordinates, reduction of order via group invariants and finding another solution from a known one via the group transformations.  
A one parameter Lie Group of transformation may be employed to reduce the order of equation by one. For second order equations, if there exists two or more symmetries, it is always possible to reduce the order of equation by two thereby obtaining full reduction. This is because any two dimensional Lie algebra is solvable and the requirement for successive reductions is the solvability of the Lie algebras. For higher order equations, if there are r symmetries and the order is k, reduction to order k-r is only possible if a solvable Lie algebra can be constructed.   
5.1. FIRST ORDER DIFFERENTIAL EQUATIONS 
For first order differential equations, the relationship of symmetries with the integrating factors will be outlined first. Group invariant solutions, reduction of order via canonical coordinates will be discussed next. Finally, mapping a solution from a known one will be treated.