By L Dresner
Lie's workforce conception of differential equations unifies the numerous advert hoc tools recognized for fixing differential equations and offers strong new how you can locate ideas. the speculation has purposes to either traditional and partial differential equations and isn't limited to linear equations. functions of Lie's concept of normal and Partial Differential Equations presents a concise, uncomplicated creation to the applying of Lie's conception to the answer of differential equations. the writer emphasizes readability and immediacy of realizing instead of encyclopedic completeness, rigor, and generality. this allows readers to speedy snatch the necessities and begin utilizing the the right way to locate options. The booklet comprises labored examples and difficulties from a variety of medical and engineering fields.
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Additional info for Applications of Lie's Theory of Ordinary and Partial Differential Equations
Once j(0) has been determined, we have two boundary conditions at x = 0 and so have converted the original two-point boundary-value problem to an initial-value problem.. 4 Singularities of the Associated Differential Equation In the example of the Thomas-Ferrni equation, Eq. 12). we have seen how the singularities of the associated differential equation played an important role. In this section, we prove two related theorems that apply when the singularities are saddle points. These theorems describe certain useful properties of the solutions y ( x ) whose images in the (p, q)-plane are the separatrices passing through the saddle points.
24 First- Orde r Ordinary Differential Equations for F and G, we can make a change of variables that will cause our differential equation to become separable. 1). 4) for which 6 = 1 and q = 1. The characteristic equations corresponding to Eqs. 7b) The difference y - x is an integral of Eq. 7~)so F can be any arbitrary function of y - x . Two independent integrals of Eq. 76) are G - y and y - x so that G equals y plus an arbitrary function of y - x. Thus one satisfactory choice of new variables g and y is From Eqs.
A) Show that p = ex is an integrating factor for the differential equation Hint: consider the partial differential equation (pM), = (PN),~. (b) Find the y-coefficient q of the infinitesimal transformation of a group G that leaves the differential equation invariant and for which the x-coefficient 6 = 0. (c) Determine the transformation equations of the group G. 1 2 The integral curves of the Clairaut equation y = x j ( 1 y2)1/2form a family of straight lines that define an envelope. (a) Show that this differential equation is invariant to the rotation group 6 = y, = -x.