Partial Differential Equations I: Basics and Separable ...

18 Partial Differential Equations I: Basics and Separable SolutionsWe now turn our attention to Differential Equations in whichthe unknown function to be deter-mined which we will usually denote byu depends on two or more variables. Hence thederivatives are Partial derivatives with respect to the various variables.(By the way, it may be a good idea to quickly review theA Brief Review of ElementaryOrdinary Differential Equations , Appendex A of these notes. We will be using some of thematerial discussed there.) Intro and ExamplesSimple ExamplesIf we have a horizontally stretched string vibrating up and down, letu(x,t)=the vertical position at timetof the bit of string at horizontal positionx,and make some almost reasonable assumptions regarding the string, the universe and the laws ofphysics, then we can show thatu(x,t)satisfies 2u t2 c2 2u x2=0wherecis some positive constant dependent on the physical properties of the stretched equation is called theone-dimensional wave equation(with no external forces).

Mar 08, 2014 · Ordinary Differential Equations, Appendex A of these notes. We will be using some of the material discussed there.) 18.1 Intro and Examples Simple Examples If we have a horizontally stretched string vibrating up and down, let u(x,t) = the vertical position at time t of the bit of string at horizontal position x ,

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