AF: Small: Solving Linear Differential in Terms of Special Functions

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Differential equations have numerous applications in science and engineering. In this project, new algorithms will be developed to compute exact solutions of linear differential equations. The project consists of two components, a top-down approach and a bottom-up approach. The bottom-up approach is to develop an algorithm for each particular type of solutions. In contrast, the top-down algorithms are not specific to any particular type of solution, instead, the top-down algorithms aim to reduce an equation to one that is easier to solve. This complements the bottom-up approach in several ways. By reducing to an easier equation, the top-down approach can drastically reduce the computation time it takes to solve the equation. Moreover, the top-down approach also reduces the number of algorithms need to be developed and implemented in the bottom-up approach; only equations that can not be reduced any further need to be treated.