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L13 - Fall 2003 Math 308/501502 1 Introduction 1.3...

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Fall 2003 Math 308/501–502 1 Introduction 1.3 Direction Fields Fri, 05/Sep c 2003, Art Belmonte Summary A first-order differential equation in normal form is written dy / dt = y 0 = f ( t , y ) . Here t is the independent variable (think of time) and y is the unknown function or dependent variable . (Recall that other letters may be used for independent and dependent variables.) A general solution in this instance is a one-parameter family of solutions to the differential equation. The parameter may be designated as C or some other letter. The graphs of members of this family are called solution curves . If we are given an initial condition (IC) y ( t 0 ) = y 0 , then the IC together with the DE constitute an initial value problem (IVP). Substituting information from the IC into the general solution of the DE allows us to determine the parameter C and thus obtain a particular solution . One meaning of the first-order normal form y 0 = f ( t , y ) is “the slope of the tangent line at a point is given by this expression.” Do this for a bunch of lattice points in a rectangular region of the ty -plane and you obtain a direction field . This is hard to draw by
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