Section 1.1 Notes - Section 1.1: Classification of...

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Unformatted text preview: Section 1.1: Classification of Differential Equations; Their Origins and Applications Definition 1 An equation with derivatives with dependent and independent variables is a differential equation. Below are examples: dy d2 y + xy ( )2 = 0 2 dx dx Derivative identities are not considered differential equations. Definition 2 A differential equation with ordinary derivatives of one or more dependent variables with respect o on independent variable is an ordinary differential equation ⇒ ODE d4 x d2 x + 5 2 + 3x = sint dt4 dt Definition 3 A differential equation with partial derivatives of one or more dependent variables with respect to one or more independent variables is called a partial differential equation ⇒ PDE ∂v ∂v + =v ∂s ∂t ∂ 2v ∂ 2u ∂ 2u + + =0 ∂x2 ∂y 2 ∂z 2 Definition 4 The order of the highest ordered derivative in a differential equation is the order of the differential equation. 1 d3 y d2 y dy + 2 + 5 + 6y 2 = 0 3 dx dx dx The Diff Eq above has an order of 3. Definition 5 A linear ODE of order n, in the dependent variable and the independent independent variable is an equation that can be expressed in the form: dn y dn−1 y dy + a1 (x) n−1 + . . . + an−1 (x) + an (x)y = b(x) n dx dx dx Below are examples of linear ODE a0 (x) d2 y dy + 5 + 3y = 0 2 dx dx d4 y d3 y dy + x2 3 + x3 = xex 4 dx dx dx Definition 6 A nonlinear ODE is ODE that is not linear d2 y dy + 5 + 6y 2 = 0 2 dx dx d2 y dy + 5( )3 + 6y = 0 2 dx dx dy d2 y + 5y + 6y = 0 2 dx dx Linear ODE are further classified by the nature of coefficients of dependent variables and derivatives The equation below is linear with constant coefficients: d2 y dy + 5 + 6y = 0 2 dx dx The equation below is linear with variable coefficients: d4 y d3 y dy + x2 3 + x3 = xex 4 dx dx dx There are many Differential Equation problems in science and engineering 2 ...
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Section 1.1 Notes - Section 1.1: Classification of...

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