By C.E. Weatherburn
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Thus each curve belongs to a one-parameter subfamily, the curves of which map into one another under the one-parameter infinitude of transformations of the group. Each such one-parameter family of curves defines a surface in (x, y , u)-space, and from the manner of its definition we see that each such surface maps into itself under the transformations of the group. 36 Second-Order Ordinary Differential Equations These invariant surfaces form a one-parameter family which we denote by the equation The invariance of each individual surface of this family means that 0 = @(x',y'.
Again we seek solutions positive on the half-line x 2 0 that vanish at infinity. 1) is invariant to the stretching group so that p > 0. Thus the solutions we seek cannot vanish at infinity proportionally to xP. Furthermore, the power-law solution has A = - 1 / 6 and thus cannot represent the asymptotic limit of a positive solution. We can find the solutions we seek from an analysis of the direction field of the associated differential equation in the ( p , 9)plane. 1 shows the fourth quadrant of the direction field of Eq.
1 ) . Since any image of the point ( 0 , y l ( 0 ) ) is ( 0 . A P y l ( 0 ) ) , we can always find a value of A for which APy1 ( 0 ) = y z ( 0 ) . Furthermore, since the point x = oo transforms into the point x' = oo, we see that y 2 ( o o ) = A B y l ( o o ) = 0. Hence the solution y , ( x ) obeying the boundary conditions y 2 ( 0 ) = a2 and ~ ~ ( 0=0 0) is an image of the solution y , ( x ) obeying the boundary conditions y1(0) = a1 and y l ( o o ) = 0 , the value of A corresponding to the transformation being ( ~ ~ / a ~ ) ~ / ~ .