MATH 255 Lecture Notes - Lecture 3: Phase Portrait, Quadratic Function, Mechanical Equilibrium
Document Summary
The methodology we have learned in nding solutions of the rst-order equa- tions in this chapter is through integration. To elaborate on this, we like to extend this approach to wider class of rst-order equations. M (t, y) + n (t, y) dy dt. = 0, where m and n are given functions. If this equation can be put into the form d dt ( (t, y(t)) = 0, then we can integrate (2) to obtain. (t, y(t)) = c. d dt ( (t, y(t)) = It is natural to match (1) and (4), i. e. , = n (t, y). (1) (2) (3) (4) (5) That is, if (5) holds, then (1) can be put into (2), and the di erential equation can be solved (in the sense that one gets rid of the derivative in (1) or (2). ) There exists a function (t, y) such that m (t, y) = . T (t, y) for all (t, y) r.
