Applications of Lie’s Theory of Ordinary and Partial by L Dresner

By L Dresner

Lie's crew thought of differential equations unifies the numerous advert hoc tools recognized for fixing differential equations and offers robust new how you can locate strategies. the idea has purposes to either traditional and partial differential equations and isn't limited to linear equations. functions of Lie's idea of standard and Partial Differential Equations offers a concise, uncomplicated advent to the applying of Lie's idea to the answer of differential equations. the writer emphasizes readability and immediacy of figuring out instead of encyclopedic completeness, rigor, and generality. this permits readers to fast grab the necessities and begin employing the how to locate suggestions. The ebook contains labored examples and difficulties from quite a lot of clinical and engineering fields.

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Example text

V) = 0 will reduce to a first-order differential equation in p and q. If the (p, q) differential equation, henceforth called the associated differential equation, can be solved explicitly, we can Lie's Reduction Theorem 37 obtain an explicit representation of the function G in Eq. 4). Since the first differential invariant q involves the variable _v, Eq. 4) is, in fact, a first-order differential equation for y in terms of x. Furthermore, since both p and q are group invariants, Eq. 4) is invariant to the group (~, TJ).

14c). 1. The curves on which dq jdp = 0 or dq jdp = ±oo divide the plane into regions in which the algebraic sign of dq jdp is constant. The locus of zero slope C 0 is the curve 4q + p3/ 2 = 0 and the locus of infinite slope Coo is the line 3 p + q = 0. The intersections of these loci are the singular points of the differential equation. 1. 14c). C0 is the locus of zero slope and Cx is the locus of infinite slope. point at P: (144, -432). In the lenticular region between these two loci, the slope dq jdp < 0; to either side of this region dq jdp > 0 as shown in the diagram.

2) dx Two integrals of Eqs. = y Q(y) P(x) Q equation F(f dxP where F ts an arbitrary function. 4) Integration of Eqs. 5b) Q(t) =A The following are some interesting special cases of these fonnulas: Case 1: P =a, Q = b, where a and bare constants. Then y) = G(cx- y) (X b y. l y' = y +b).. 6c) and F and G are arbitrary functions. translation group. The group is a pure P =ax, Q =by, where a and b are constants. 7a) x' = xea). lX y' = yeh). l = ea). 7c) This group is a pure stretching group. Case 3: P = a I x, Q = by, where a and b are constants.

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