Chapter 1 Slides

# Chapter 1 Slides - DIFFERENTIAL EQUATIONS Basic Terminology...

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DIFFERENTIAL EQUATIONS Basic Terminology A diﬀerential equation is an equa- tion that contains an unknown func- tion together with one or more of its derivatives. 1

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Examples 1. y ± =2 x + cos x 2. dy dt = ky 3. x 2 y ±± - 2 xy ± +2 y =4 x 3 4. 2 u ∂x 2 + 2 u ∂y 2 =0 5. d 3 y dx 3 - 4 d 2 y dx 2 +4 dy dx 2
TYPE: If the unknown function depends on a single independent variable, then the equation is an ordinary diﬀer- ential equation ; if the unknown function depends on more than one independent variable, then the equa- tion is a partial diﬀerential equa- tion . 3

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ORDER: The order of a diﬀerential equation is the order of the highest derivative of the unknown function appearing in the equation. 4
Examples: 1. dy dt = ky 2. x 2 y ±± - 2 xy ± +2 y =4 x 3 3. 2 u ∂x 2 + 2 u ∂y 2 =0 4. d 3 y dx 3 - 4 d 2 y dx 2 +4 dy dx 5. y ±± x sin y ± +3 e xy = d 3 dx 3 ( e 2 x ) 5

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SOLUTION: A solution of a diﬀerential equa- tion is a function deFned on some interval I (in the case of an or- dinary diﬀerential equation) or on some domain D in two or higher dimensional space (in the case of a partial diﬀerential equation) with the property that the equation re- duces to an identity when the func- tion is substituted into the equation. 6
Examples: 1. y ± =2 x + cos x y = x 2 + sin x , y = x 2 + sin x + C 2. y ± = ky y = e kt ,y = Ce 7

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3. x 2 y ±± - 2 xy ± +2 y =4 x 3 y = x 2 x 3 ; y =2 x + x 2 not a solution.
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Chapter 1 Slides - DIFFERENTIAL EQUATIONS Basic Terminology...

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