By L Dresner
This creation to the applying of Lie's conception to the answer of differential equations comprises labored examples and difficulties. The textual content exhibits how Lie's crew thought of differential equations has purposes to either usual and partial differential equations.
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Extra resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
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 ~ ) ~ / ~ .