Ch05 Transient Analysis 1s09 - Chapter 5 Differential...

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Introduction to Differential Introduction to Differential Equations Equations Transient Analysis of Transient Analysis of First First -Order Networks Order Networks Chapter 5 Department of Electrical and Electronics Engineering University of the Philippines - Diliman Department of Electrical and Electronics Engineering EEE 33 - p2 Differential Equations Definition : Differential equations are equations that involve dependent variables and their derivatives with respect to the independent variables. 0 2 2 = + ku dx u d Simple harmonic motion: u(x) 2 2 2 2 2 2 2 2 2 t u c z u y u x u = + + Wave equation in three dimensions: u(x,y,z,t) Department of Electrical and Electronics Engineering EEE 33 - p3 Ordinary Differential Equations Definition : Ordinary differential equations (ODE) are differential equations that involve only ONE independent variable. 0 2 2 = + ku dx ) x ( u d u(x) is the dependent variable x is the independent variable Example: Department of Electrical and Electronics Engineering EEE 33 - p4 Ordinary Differential Equations We can classify all ODEs according to order, linearity and homogeneity . The order of a differential equation is just the highest differential term involved: 2 nd order 3 3 dt x d x dt dx = 0 0 1 2 2 2 = + + a dt dy a dt y d a 3 rd order Department of Electrical and Electronics Engineering EEE 33 - p5 Linearity The important issue is how the unknown variable (ie y ) appears in the equation. A linear equation must have constant coefficients , or coefficients which depend on the independent variable . If y or its derivatives appear in the coefficient the equation is non-linear. is linear 0 = + y dt dy 0 2 = + x dt dx is non-linear is linear 0 2 = + t dt dy 0 2 = + t dt dy y is non-linear Department of Electrical and Electronics Engineering EEE 33 - p6 Linearity - Summary y 2 dt dy dt dy y dt dy t 2 dt dy y t ) sin 3 2 ( + y y ) 3 2 ( 2 - Non-linear Linear 2 y ) sin( y or
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Department of Electrical and Electronics Engineering EEE 33 - p7 Homogeniety Put all the terms of the differential equation which involve the dependent variable on the left hand side (LHS) of the equation. Homogeneous : If there is nothing left on the right-hand side (RHS), the equation is homogeneous. (unforced or free) Nonhomogeneous : If there are terms left on the RHS involving constants or the independent variable, the equation is nonhomogeneous (forced) Department of Electrical and Electronics Engineering EEE 33 - p8 Examples of Classification 0 = + y dx dy s 1 st Order s Linear s Homogeneous ) sin( ) cos( 2 2 2 x y x dx y d = + s 2 nd Order s Non-linear s Non-homogeneous ) x cos( y dx y d = - 4 5 3 3 s 3 rd Order s Linear s Non-homogeneous Department of Electrical and Electronics Engineering EEE 33 - p9 Linear Differential Equations A linear ordinary differential equation describing linear electric circuits is of the form ) t ( v a dt dx a ... dt
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Ch05 Transient Analysis 1s09 - Chapter 5 Differential...

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