By L Dresner
This creation to the appliance of Lie's idea to the answer of differential equations contains labored examples and difficulties. The textual content indicates how Lie's crew concept of differential equations has functions 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
Since stretching groups occur frequently in practice, a detailed discussion is warranted. The most general stretching group in two variables x and y where is a constant. No generality is lost by choosing the exponent of h in Eq. 1a ) to be 1. 1) is an arbitrary 40 Second-Order Ordinary Diferential Equations function of three independent integrals of the characteristic equations ( 3 . 1 . 2 b ) ,which now take the form The functions y / x P , j,/xP-l and j;/xP-? are three such integrals, so the most general second-order differential equation invariant to the group ( 3 .
Show, therefore, that Lie's integrating factor for this differential equation is y = (M2 N ')-'I2. (According to Cohen [Co-1 I , p. ) (d) The converse of part (c) is also true, namely, if y = ( M ~ N2)-'I2 is an integrating factor of the differential + + + 3 + + 31 Tabulation of Differential Equations + equation M (x. y) dx N (x, y ) dy = 0, then the integral curves of this differential equation are parallel to one another. Can you prove this result, too? (a) Show that p = ex is an integrating factor for the differential equation Hint: consider the partial differential equation (pM), = (PN),~.
No generality is lost by choosing the exponent of h in Eq. 1a ) to be 1. 1) is an arbitrary 40 Second-Order Ordinary Diferential Equations function of three independent integrals of the characteristic equations ( 3 . 1 . 2 b ) ,which now take the form The functions y / x P , j,/xP-l and j;/xP-? are three such integrals, so the most general second-order differential equation invariant to the group ( 3 . 3 . 1 ) must have the form where @ can be any function. From Eq. ( 3 . 3 . 4 ) it follows that there are real power-law solutions of the form y = A x @ when A is a real constant that satisfies the algebraic equation Under circumstances outlined below, one of them gives the asymptotic behavior of positive solutions on the half-line x 2 0 that vanish at infinity.
Applications of Lie's Theory of Ordinary and Partial Differential Equations by L Dresner
Categories: Mathematical Physics