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Math 230 Sec 1.1

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Differential Equations Notes
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  1 Math 230 – Section 1.1 Basic definitions and terminology I.   Differential equations Definition: A differential equation  (DE) is an equation containing the derivatives or differentials of one or more dependent variables, with respect to one or more independent variables. Ex: Newton’s equation     2  2 =   (  ,  )  Classification Type: : by type, order, and linearity 1.   Ordinary differential equation (ODE): contains only ordinary  derivatives of one or more dependent variables with respect to a single  independent variable. Ex : 2.   Partial differential equation (PDE): contains partial  derivatives of one or more dependent variables with respect to 2 or more  independent variables. Exa)   Laplace equation :  b)   Heat equation c)   Wave equation  2 Order: The order of a DE is the highest derivative’s order that appears in the equation. Ex : The general form for the n-th order ODE is � ,  ,  ,  2  2 , ⋯ ,      =0   Linearity: 1.   Linear: Ex : The general form for a linear ODE is   (  )     +  − 1 (  )  − 1  − 1 + ⋯ +  2 (  )  2  2 +  1 (  )  +  0 (  )  =   (  )  2.    Nonlinear: Ex :  3 II.   Solution of a differential equation Definition: A function    defined on an interval   is said to be a solution of a DE if it reduces the equation to an identity when substituted into the equation. Ex : verify that the indicated function is a solution of the given differential equation. a)  +20  =24;  =65 − 65  − 20    b)  ′′ +  =sec  ;  =cos  ln(cos  )+  sin   for 0<   <  2    4 c) 2     +(  2 +2  )  =0;  2  +  2 =    where   is a constant.  Definitions1.   Trivial solution: : 2.   Explicit solution: 3.   Implicit solution: n -parameter family of solutions: ExConsider the ODE  ′ =   1/2  : one-parameter family of solutions
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