ABSTRACT
Differential equations model many physical situations, and the form of a solution depends on the structure of the differential equation involved. This chapter shows how a general solution to a differential equation must contain arbitrary constants of integration equal in number to the order of the equation – that is, to the order of the highest derivative involved. The notions of direction fields and isoclines for first-order equations are introduced and used to provide qualitative information about the set of all solutions to a differential equation once its form has been specified. All linear differential equations are characterized by the fact that the dependent variable and its derivatives only occur with degree 1, while the coefficients multiplying them are either constants or functions of the independent variable. The chapter considers a number of essentially different physical problems and in each case takes the discussion as far as the derivation of the governing differential equation.
