A stepwise Cas course for solving First-Order Partial Diferential Equations

Abstract

Partial Differential Equations (PDE) is a very important topic in advance Mathematics for Engineering. The three main first-order PDE problems that a basic course must deal with are:

\begin{enumerate} \item {\bf Pfaff Differential Equations}, which consists on finding the general solution for: $$ P(x,y,z)\:{\rm d}x + Q(x,y,z)\:{\rm d}y + Q(x,y,z)\:{\rm d}z = 0 $$ \item {\bf Quasi-linear Partial Differential Equations}, which consists on finding the general solution for: $$ P(x,y,z)\:p + Q(x,y,z)\:q = R(x,y,z) $$ where $\displaystyle \quad p=\frac{\partial\: z}{\partial\: x}\quad$ and $\displaystyle \quad q=\frac{\partial\: z}{\partial\: y}$. \item {\bf Lagrange-Charpit Method} for finding a {\em complete integral} for a given general first order partial differential equation: $\quad F(x,y,z,p,q)=0$. \end{enumerate}

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In order to help the teaching and learning process of this topic, we have developed, using {\sc Derive} 6 {\sc Cas}, the file \textbf{\tt FOPDE.mth}. The use of this file allows the user to solve these three problems (their general cases and the particular cases) stepwise. Since these three problems requires several steps for their resolution, the programs developed in {\tt FOPDE.mth} show step by step all the resolution procedure providing in this way a powerful tool as a tutorial for teaching how to solve these types of equations. This way the use of {\sc Derive} 6 is done as a {\sc PeCas} (Pedagogical {\sc Cas}) providing not only the final result but also all partial results.

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On the other hand, in the resolution process of such equations, first-order Ordinary Differential Equations (ODEs) are needed. Therefore, {\tt FOPDE.mth} loads the package {\tt FOODE.mth}, which is part of the {\sc Derive} package that was introduced in TIME 2010.

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The programs contained in the file \textbf{\tt FOPDE.mth} can be grouped within the following blocks: \begin{itemize} \item \textbf{First-order ODEs}: separable equations and equations reducible to them, homogeneous equations and equations reducible to them, exact differential equations and equations reducible to them (integrating factor technique), linear equations, the Bernoulli equation, the Riccati equation. \item \textbf{First-order differential equations and nth degree in y'}. \item \textbf{Generic programs to solve first order differential equations}. \item \textbf{Pfaff Differential Equations}. \item \textbf{Quasi-linear PDE}. \item \textbf{Lagrange-Charpit Method for First-Order PDE}. \end{itemize}

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Finally, we will state the conclusions obtained after using this file with our students and also some future work on this and other related subjects.