Differential Equations the Basic Reduction to First Order Systems

2 Differential Equations the Basic Reduction to First Order Systems 

Scalar equations, as studied earlier, depend on one or several real variables x, y, t, T,. Differential equations instead link various derivatives of one or more functions /(...), y(...). each with any number of variables. These mathematical functions describe state variables in engineering parlance. Differential equations are equations in one or more variables and in one or more functions of the variables and in their derivatives. They involve independent variables such as space and time, and dependent, so called state variables or functions and their derivatives. Many physico-chemical processes are governed by differential equations or systems thereof, that involve unknown functions/, g,. .. in various variables and various of their derivatives/, g1, g ,. .. 

The task of modeling is to obtain valid scalar, differential, or other type of equations (integral, integro-differential, etc) that describe a given physical system accurately and efficiently in mathematical terms. Numerical analysis and computations then lead us to the solution values or to the solution function(s) themselves from the model equations. 

A different classification scheme for DEs, short for differential equations, separates those DEs with a single independent variable dependence, such as only time or only 1-dimensional position, from those depending on several variables, such as time and spatial position. DEs involving a single independent variable are routinely called ODEs, or ordinary differential equations. DEs involving several independent variables such as space and time are called PDEs, or partial differential equations because they involve partial derivatives. 

As this book is mainly intended for undergraduates, we only treat those chemical/biological processes and problems that can be modeled with ODEs. PDEs are significantly more complicated to understand and solve. However, we will often have to solve systems of ODEs, rather than one single ODE. Such ODE systems contain several DEs in one and the same independent variable, but they generally involve several functions or state variables in their formulation. In fact, systems of ODEs (and matrix DEs) occur quite naturally in chemical/biological engineering problems. 

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