By L Dresner
This creation to the applying of Lie's concept to the answer of differential equations contains labored examples and difficulties. The textual content indicates how Lie's crew conception of differential equations has purposes to either usual 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.
Y) for which y < x + 1/4, one at points for which y = x + 1/4, and none at other points. It can be shown by direct substitution that the one-parameter family where c is a constant, -00 < c < m , satisfies Eq. 1 ). 2), points for which y = x 1 /4 lie on one curve, and other points on none. 1. y =x 21 Curves of the family y = c - (c - x12 and their envelope + 144. + Thus the form +(x. 1). 2) has the envelope The envelope of a family of curves is a curve that is tangent at each of its points to a member of the family.
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. rOthe solution is negative and thus unacceptable. Hence, we have found solutions to Eq. 1) that are positive on the interval (0. ro. 2). Hence, to find them all, we have to integrate Eq. I ) numerically only once. Now the point P in the (p. r = 0 means moving along the separatrix S away from P in the direction of positive p .
Applications of Lie's Theory of Ordinary and Partial Differential Equations by L Dresner