By L Dresner
This creation to the applying of Lie's concept 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
1. The fourth quadrant of the direction field of differential . is the locus of zero slope and C , is the locus of infinite equation ( 3 . 5 . 3 ~ )Co slope. are minima. Positive solutions that vanish at infinity cannot have only minima and hence must be monotone decreasing. ) The locus of zero slope Co is given by the equation The locus of infinite slope C, has two branches, one in the first and third quadrants due to the vanishing of q - 2 p and 52 Sucontl-Ortlrr Ordincrn Diflrrential Eqrtutiorts a second, the q-axis, at which the function f ( q .
U'. = $(X(X,y : A). Y(x, y : A). U(x. Y , A). 2) If we differentiate Eq. @, olh =0 the characteristic equations of which are C) + 11; + If p(x. y) and q(x, y . u ) are two integrals of Eqs. 3b), the most general solution for @ is an arbitrary function G of p and q. 1) can be represented by the equation The function p(x, y), being an integral of the first pair of Eqs. 3b), is a group invariant. The function q(x, y . u ) = q(x. y. y), which is an invariant of the once-extended group, is called a jirst difierential invariant.
1 . 1 ) are explicitly known, it is possible to calculate the invariant p by algebraic manipulation and the first differential invariant q by differentiation and algebraic manipulation. This is proved in Appendix D. Example: The Emden-Fowler equation arises in the study of the equilibrium mass distribution in a gas cloud held together by gravitation; the exponent n is related to the adiabatic exponent y of the gas by y = (n I)/n. 5) is invariant to the twice-extended stretching BrouP + y' = ABy j,' = AB-lj, ' A 8-2 y..