By L Dresner
This creation to the appliance of Lie's thought to the answer of differential equations comprises labored examples and difficulties. The textual content indicates how Lie's staff thought of differential equations has functions to either usual and partial differential equations.
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Extra resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
Sample text
3), find a substitution for u that separates the variables. Thus find a substitution for y that separates the variables in the Bernoulli equation. (b) Use Eqs. 6) to calculate 6 and q for a group that leaves the Bernoulli equation invariant. Tubulution of Differential Eqrtutions 33 (c) Determine relations that the coefficients 6 and q must obey directly from the Bernoulli equation and use them to check your results from part (b). 1 7 Find a group which leaves invariant both the integral curve y = coex of the elementary differential equation j = y and the differential equation itself.
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. Then, since v/x" y1/x@, -+ which completes the proof. rample: The differential equation arises in the study of heat transfer in superfluid helium. 1) with = -2. Substitution of the form y = AX-' into the differential equation yields A = 4/(3 4 3 ) so that y, = ( 4 / [ 3 , / 3 ] ) ~ - ~ . To show that the positive solutions for which y(oo) = 0 are ordered, we set u = 8 y , the difference between two infinitesimally close neighboring solutions.
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. 5. I . e. if the transformation of x by the group is a stretching, then 6 = x and we can satisfy Eq. 14~)by choosing F = x. The function G is a group invariant (see Eq. 2)). Thus for groups in which the transformation of x is a stretching, introducing a group invariant as a new variable in place of y and leaving the variable x unchanged results in a new differential equation that is separable.