By L Dresner
This advent to the applying of Lie's thought to the answer of differential equations comprises labored examples and difficulties. The textual content indicates how Lie's staff conception of differential equations has purposes to either traditional and partial differential equations.
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Additional resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
2) to find the transformation equations of two different groups to which this differential equation is invariant. 4 Find a group under which the differential equation Y = ax + by c is invariant, calculate Lie's integrating factor and find the integral curves of the differential equation. ; = (In y-ex) exp(x+er) is invariant to the group x' = In($ A), y' = Ay. 5) to determine a change of variables that separates the variables and find the form the differential equation takes in the new variables.
The analysis given below suggests a way to determine this integral curve. The method employed applies to many applications including the one described below. 14c), like that of any first-order differential equation, is a one-parameter family of curves @ ( p q, ) = c. 13) for which y (oo) = 0; for any one of these curves y(x) is uniquely determined by its value y(0) at x = 0. An image yt(x') of y(x) has the value A - ' ~ ( 0 ) at x' = 0 and also vanishes at x' = oo. 1). Now since xt3y'= x'y and ~ ' ~ j=l x4jl, y'(xt) of p(x) determines the same locus in the (p, 9)-plane as the curve y(x) itself.
For the group permutes the curves of the family among themselves and thus leaves the family as a whole unchanged; therefore it leaves the envelope of the family unchanged. An envelope, if it exists, is thus invariant to all the groups that leave the differential equation invariant. 5 Change of Variables Lie proposed a second method of solving first-order differential equations that are invariant to a group, which involves using the group to find new variables ~ ( xy). and y(x, y), in terms of which the differential equation becomes separable.
Applications of Lie's Theory of Ordinary and Partial Differential Equations by L Dresner