First-Order Differential Equations

Most differential equations cannot be solved explicitly by “closed formulas,” that is, formulas which consist of a finite number of known functions and their integrals. Nevertheless, we start the study of differential equations with several solution methods for some well-known, useful equations of the first order. We introduce linear equations, homogeneous and nonhomogeneous, separable equations, equations that can be solved by suitable substitutions, exact equations, and integrating factors. Second-order equations of the form y ″ = f ( y ) $$y''=f(y)$$ are discussed here since they also lead to equations of the first order. We introduce some concepts that will accompany us later on, as general solution vs. solution of an initial value problem, explicit vs. implicit solutions, singular solutions, etc. We emphasize the subtle points, questions which seem to have self-evident answers but hide deep ideas. Finally, we discuss geometric ideas like direction fields, isoclines, orthogonal solutions, and how to use them for the equations that we met before.