Classification of Ordinary Differential Equations

The concentrations of liquid and vapor are related by the equilibrium relationship 

If the assumptions of constant molar overflow and negligible delay in vapor flow are made, then V I = V . The delay in liquid flow is 

The above equations applied to each stage in a multistage separation process result in a large set of simultaneous ordinary differential equations. 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

Ordinary differential equations are classified according to their order, their linearity, and their boundary conditions. 

---