By Derek F. Lawden

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**Extra info for A Course in Applied Mathematics, Vol.1, 2**

**Sample text**

1J ), Lie has shown how to reduce the problem of solving it to that of solving two first-order differential equations one at a time successively. Lie's way of doing this is explained in the next section. 3b, c) determine a two-parameter family of curves in the three-dimensional space whose coordinates are x. y and u. (Consider the direction field of infinitesimal vectors dx, dy = y dx, du = udx determined by solving Eqs. Y and u. 3b, c) are invariant to the once-extended group (~, 1], 1Jt}, the transformations of the group carry each of these curves into other curves of the family.

8) which completes the proof. 9) arises in the study of heat transfer in superftuid helium. 1) with f3 = -2. Substitution of the form y = Ax- 2 into the differential equation yields A = 41(3-j3) so that y* = (41[3-J3])x- 2. To show that the positive solutions for which y( oo) = 0 are ordered, we set u =

Thus as x ~ oo, a' ~ oo (remember f3 < 0). This means as the point (x, y) moves out along the solution S toward x = oo, the image point (x', y') moves steadily upwards along the vertical line x' = c. 8) which completes the proof. 9) arises in the study of heat transfer in superftuid helium. 1) with f3 = -2. Substitution of the form y = Ax- 2 into the differential equation yields A = 41(3-j3) so that y* = (41[3-J3])x- 2. To show that the positive solutions for which y( oo) = 0 are ordered, we set u =