Separable Differential Equations in Mathematics
Discussing separable differential equations in mathematics, covering topics such as order, degree, separation of variables, and examples to understand the process of finding solutions. Learning outcomes include familiarity with separable differential equations and their solutions.
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Kurdistan Region Iraq Ministry of Higher Education and Scientific Research Salahaddin University-Erbil College of Science Department of Mathematics ??? Stage ???????? ???????????? ????????? Lecture 2 Separable Differential Equations By: MSc. Hersh M. Saber E-mail: hershmwhamad@yahoo.com 2019-2020
Previous Lecture Solutions of differential equations. Order and degree. 1. Find order and degree of ?2? ??2+ sin ? 2 = ??3? ?? ?? ?? ??= 1. ??3 , b) ? a) 1 +
Outline In this lecture, we will cover the following topics: 2.1 Separable differential equations. 2.2 Examples.
Learning outcome We will get familiar with separable differential equations and identify it. We will learn the process for finding solution of separable differential equations.
2.1 Separation of variables (Separable differential equations) The first order differential equation of the form ? =?? ??= ? ?,? ..(1) If we may be able to factor ?(?,?) into factors containing only ? or ?, but not both, then we say ?(?,?) is separable. Thus; ? ?,? = ? ? ? ? .
2.1 Separation of variables (Separable differential equations) When the variables are separable, then differential equation (1) becomes ?? ??= ? ? ? ? ? ?= ?(?)??. .(2) ??
2.1 Separation of variables (Separable differential equations) then integration of both sides of equation (2), we have 1 ? ??? = ? ? ?? + ?. where C is a constant.
2.1 Examples Remark 1. It is possibility that either ? or ? may be constant function. Example 1. Identify the separable equations: a. ? = ?2?2 b. ? =? ? d. 2? 1 ?2 1 ? + ? ? 1 + ?? = 0 ?+? c. ? = ?(?) Solution: Equations (a), (c) and (d) are separable. Equation (b) is not separable.
2.1 Examples Example 2. Solve the differential equations: a. ? = 3?2? ??.? = 2 ? + ?2? Solution: (a) ? = ln(?3+ ?), which is the general solution where ? is an arbitrary essential constant. (b) ? = tan ?2 + ? , is the general solution where ? is an arbitrary essential constant
Homework Homework 1. Identify the separable equations from among the following list of differential equations: ? = ? ? ? , b. ? =? ? , c. ? = ? ?2. a. Homework 2. Solve the following differential equations: a. ??? ??? = ????, b. ?21 ? ?? + ?2(1 + ?)?? = 0.
Next Lecture Homogeneous differential equations and Examples.
References [1] Polyanin, Andrei D., and Valentin F. Zaitsev. Handbook of ordinary differential equations. CRC Press, 2017 .[2] William A. Adkins, and Mark G. Davidson. Ordinary Differential Equations Undergraduate Texts in Mathematics. Springer Science+Business Media New York 2012 11
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