By L Dresner
This creation to the appliance of Lie's concept to the answer of differential equations comprises labored examples and difficulties. The textual content exhibits how Lie's team thought of differential equations has functions to either traditional and partial differential equations.
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Additional resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
Thus the singular point P: (0. 4. Now if we approach the point P along the line then it follows from Eq. ( 3 . 5 . 3 ~that ) to leading order as long as m # -3/2. r = s o exp L Z ; 3 ) If we apply l'H6pital's rule to the right-hand side of Eq. 3~). we find that the separatrix S has the slope -3/4 at P. r = -ro, a constant, at P. rO) = 0. xO)/~xO = qp = -2/3 so that j ( x 0 ) = -2xo/3. These two conditions provide initial conditions for a huckkrcrrd integration of Eq. r = 0. Because the slope j(xo) is negative, a backward integration produces a positive solution on the interval (0.
The locus of zero slope Co is given by the equation The locus of infinite slope C, has two branches, one in the first and third quadrants due to the vanishing of q - 2 p and 52 Sucontl-Ortlrr Ordincrn Diflrrential Eqrtutiorts a second, the q-axis, at which the function f ( q . p) becomes infinite. Thus the singular point P: (0. 4. Now if we approach the point P along the line then it follows from Eq. ( 3 . 5 . 3 ~that ) to leading order as long as m # -3/2. r = s o exp L Z ; 3 ) If we apply l'H6pital's rule to the right-hand side of Eq.
If U ( X ) were negative anywhere, it would have to have a negative minimum somewhere since it starts out positive and is zero at infinity. But at a negative minimum, u < 0 and ii 2 0. Since ( 3 / 2 ) ~ - ' / ~ yI ,' 0, / ~ these conditions violate Eq. 13). Thus the assumption that u ( x ) is negative leads to a contradiction. This is enough to prove the ordering. 14~)cannot be solved explicitly, so how shall we proceed? This is an important question because in many applications of Lie's reduction theorem the associated differential equation cannot be integrated in closed form.