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**Unformatted text preview: **Monday, January 23 − Lecture 10 : Differential equations: Definitions − Separable DE’s. (Refers to Sections 10.1 and 10.2 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Recognize a solution to a differential equation and an initial value problem, recognize when a DE is linear and when a DE is non-linear, recognize separable DE’s, solve a first order separable DE, solve an IVP. 10.1 Definition − A differential equation is an equation containing at least one derivative of a variable. The order of a differential equation is the highest order of a derivative which appears in the equation. So a first-order differentiation equation is one where the highest order of the derivative is 1. A differential equation of order 1: A differential equation of order 2: We often use the acronym DE as an abbreviation of the words “differential equation”. Sometimes we use ODE, which stands for “ordinary differential equation” to distinguish from another type of DE called “partial DE” which we do not discuss here. To “ solve a DE ” means to express the DE in a form which is equivalent, that is, an equation which has the same solutions as the original DE, but which does not contain any derivatives. Some differential equations are easily solved since they only require straight integration. For example the differential equation (also written as or has an infinite family of solutions y = − cos x + e x + C . - The expression y = − cos x + e x + C...

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