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.

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.