Tiêu đề 1.1-1.3- Introduction to DEs .pdf ✅

1.1- 1.3- Basics of Differential Equations Learning Objectives A. Investigate the language of differential equations B. Initial-Value Problems __________________________________________________________________________________________________ A. Investigate the language of differential equations Definition: An ordinary differential equation (ODE) is *all ODEs studied in this chapter will be simply referred to as Differential Equations Classification by Order Definition: The Order of a DE is the order of the highest derivative in the equation. Ex 1) Give the order of each differential equation. �� �� = �(� − ��) � �!� ��! = � �� �� = �� � �� �� = �� − ��! �!� ��! + � � sin � = 0 �� �� + �(�)� = �(�)


Ex 3) Derive a differential equation that has (implicit) solutions �! + �! = �! for � ≥ 0. Differential equations arise in every scientific field. They are usually the byproduct of analyzing rates of change between quantities. More formally, these equations arise from mathematical or physical laws. Therefore, it is crucial to develop techniques to solve many different types of DEs. Tractrix: Suppose you have a line segment from the origin to � = �. and we drag the end at the origin along the �-axis. Find the differential equation that, when solved, would produce the path of the right end of the line segment.