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by Fritz Schwarz (Author)
Loewy's results for ordinary differential equations.- Rings of partial differential operators.- Equations with finite-dimensional solution space.- Decomposition of second-order operators.- Solving second-order equations.- Decomposition of third-order operators.- Solving third-order equations.- Summary and conclusions.- Solutions to the exercises.- Solving Riccati equations.- The method of Laplace.- Equations with Lie symmetries.
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The central subject of the book is the generalization of Loewy's decomposition - originally introduced by him for linear ordinary differential equations - to linear partial differential equations. Equations for a single function in two independent variables of order two or three are comprehensively discussed. A complete list of possible solution types is given. Various ad hoc results available in the literature are obtained algorithmically. The border of decidability for generating a Loewy decomposition are explicitly stated. The methods applied may be generalized in an obvious way to equations of higher order, in more variables or systems of such equations.
