By Dongming Wang

This e-book offers the state-of-the-art in tackling differential equations utilizing complicated tools and software program instruments of symbolic computation. It specializes in the symbolic-computational features of 3 different types of primary difficulties in differential equations: remodeling the equations, fixing the equations, and learning the constitution and homes in their strategies. The 20 chapters are written through top specialists and are established into 3 parts.

The e-book is worthy studying for researchers and scholars engaged on this interdisciplinary topic yet can also function a helpful reference for everybody drawn to differential equations, symbolic computation, and their interaction.

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**Extra resources for Differential Equations with Symbolic Computation **

**Example text**

Let us prove the necessity. If a30 = b30 = 0, then from µ15 = 0 we have a03 b212 = b03 a212 , so condition (v) holds. If a30 = 0 or b30 = 0, then by µ10 = (a12 a30 − b12 b30 )/5 there is a constant p, such that a12 = p b30 , b12 = p a30 . 6) into every expression of µk and simplifying them it is easy to complete the proof. 15]) to get the center conditions of the system, we need ﬁnd out all the elementary Lie invariants of the system. From the technique used in [7], we have the following lemma.

Using such a ˙ adek technique, Zol ¸ has shown that there are cubic systems with 11 limit cycles bifurcating from a single critical point [17]. However the proof is quite technical and in general such methods are hard to apply to systems of higher degree. Estimating Limit Cycle Bifurcations 25 Na¨ ¨ıvely, we would expect the number of limit cycles to be estimated by one less than the maximum codimension of a component of the center variety. A comparison of the cyclicities of the Li´enard systems computed in [4] with the codimensions of their center varieties, using the results of [2], shows that this is indeed the case for Lienard ´ systems of low degree.

G. Lloyd, C. Christopher, J. Devlin, J. M. Pearson, and N. Yasmin, Quadraticlike cubic systems. Planar nonlinear dynamical systems (Delft, 1995). Diﬀerential Equations Dynam. Systems 5 no. 3-4 (1997), 329–345. [9] N. G. Lloyd and J. M. Pearson, Computing centre conditions for certain cubic systems. J. Comput. Appl. Math. 40 no. 3 (1992), 323–336. [10] N. G. Lloyd and J. M. Pearson, Conditions for a centre and the bifurcation of limit cycles in a class of cubic systems. Bifurcations of planar vector ﬁelds (Luminy, 1989), Lecture Notes in Math.