By L Dresner
Lie's staff idea of differential equations unifies the various advert hoc tools recognized for fixing differential equations and gives strong new how one can locate ideas. the idea has functions to either traditional and partial differential equations and isn't limited to linear equations. functions of Lie's thought of normal and Partial Differential Equations presents a concise, uncomplicated creation to the appliance of Lie's thought 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 take hold of the necessities and begin utilizing the the way to locate options. The publication contains labored examples and difficulties from a variety of medical and engineering fields.
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Additional resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
10 A curve C2 is said to be parallel to a curve CI if the distance from C I along its normal to the curve C2 is the same for all points of C l . (a) Prove that any normal to the curve CI is also normal to the curve C2. (b) Imagine the (x, y)-plane filled with the one-parameter family F of curves parallel to Cl . Denote by Cs the curve parallel to C1 at a distance 6 from C1. Define the set of transformations T, of points P: (x, y) into images P': (x', y') in the following way. Through the point P passes a curve C, of the family F.
2 Find a group which leaves the differential equation y = xy(1 In y x2) invariant, and using Lie's integrating factor find the general solution. 3 The linear, first-order, inhomogeneous differential equation y P(x) y = Q(x) has the well-known integrating factor exp(l," P ( 2 ) dz). (Here z is a dummy variable of integration. 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.
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. 1. The curves on which dqldp = 0 or dqldp = koo divide the plane into regions in which the algebraic sign of dq/dp is constant. The locus of zero slope Co is the curve 44 p2/*= 0 and the locus of infinite slope C, is the line 3 p q = 0. The intersections of these loci are the singular points of the differential equation. 1. 14~). Co is the locus of zero slope and C, is the locus of infinite slope.