By L Dresner
Lie's staff concept of differential equations unifies the various advert hoc equipment identified for fixing differential equations and gives strong new how one can locate ideas. the speculation has purposes to either traditional and partial differential equations and isn't limited to linear equations. purposes of Lie's conception of standard and Partial Differential Equations presents a concise, uncomplicated creation to the appliance of Lie's conception to the answer of differential equations. the writer emphasizes readability and immediacy of figuring out instead of encyclopedic completeness, rigor, and generality. this allows readers to fast seize the necessities and begin making use of the tips on how to locate suggestions. The booklet comprises labored examples and difficulties from a variety of clinical and engineering fields.
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Extra info for Applications of Lie's Theory of Ordinary and Partial Differential Equations
6). 3) belongs to the integrating factor ,Y = y-' and thus should leave only the integral curve y = 0 invariant. 3).. p) = co invariant. The most general integrating factor v of Eq. 3) where p is some particular integrating factor and F is an arbitrary function of $. If , Y does not already equal zero for the integral curve, we can make v-I = 0 by choosing F so that F(c,) = W. 4 Singular Solutions General as the work of the last two sections may seem, it is nonetheless based on the tacit assumption that at each point of the (x.
Thus prove that the differential equation is invariant if its family of integral curves is an invariant family. Can your proof be inverted to prove the converse, namely, that if the differential equation is invariant then its family of integral curves is invariant? 7 Goursat gives the integrating factor v = xPy4 for the differential equation y(a a x m y n dx ) + x(b + @xmyn) dy = 0, where p and q are determined by the simultaneous equations bp - a q = a - b and @p- a q = a ( n 1 ) - p(m 1 ) [Da-621.
8a, 6) for x and y, we find, after rearrangement which is identical to Eq. 2).. Change of Variables 25 Another choice of ( and q that will cause the variables to separate is ( = g and q = 0. hen -1q = - y- and Eqs. 2) become A A The quantities gy and y are integrals of Eq. 12) and thus the most general difErenti3 equation invariant to the group ( E . q) is A which is also separable. So if we can solve Eqs. 3a, b ) , which now take the form for F and G, we can make another change of variables that will also cause our differential equation to separate.