By Peter J. Olver
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Extra resources for Applications of Lie groups to differential equations MCde
17) really hinges on the fact that we have the Lie group that leaves our differential invariant. 13) then a nonlinear PDE for and that is really too hard to solve! So at this point it seems rather hopeless. However, Lie considered not the Lie group itself but an expanded form of the Lie group which are called infinitesimal transformations. 18) about . 22 These are referred transformations and infinitesimals. to as and infinitesimal as simply Here we consider two examples where the infinitesimals are obtained from the Lie group.
In an attempt to answer this question, we consider these transformations in detail. 5 (cf. Exercise Chapter 1, #8). 5). 9). 5). 14 holds. 17b 40 two functional equations for and . 17) really hinges on the fact that we have the Lie group that leaves our differential invariant. 13) then a nonlinear PDE for and that is really too hard to solve! So at this point it seems rather hopeless. However, Lie considered not the Lie group itself but an expanded form of the Lie group which are called infinitesimal transformations.
We usually try the following forms (see the exercises): Exercises 1. For the given Lie group, find the corresponding infinitesimals and 2. For the given infinitesimals and the corresponding Lie group. 54 . , find 3. Find infinitesimal transformations leaving the following ODEs invariant. Use these to find a change of variables and reduce the original ODE to one that is separable and solve the equation. Hints: Try: a. b. c. 4. 44. 3 Standard Integration Techniques When one first learns the standard techniques for solving first-order ordinary differential equations, one creates a cookbook on how to solve a wide variety of equations: linear, Bernoulli, homogeneous, exact, Riccati, etc.
Applications of Lie groups to differential equations MCde by Peter J. Olver