M334Introduction

M334Introduction - Math 334 Dierential Equations...

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Math 334: Differential Equations Introduction What is a Differential Equation? A differential equation is an equation containing one or more of the derivatives of a function. Oftentimes, the variable assigned as the output of f appears in the differential equation in place of f as well as in the derivatives of f : for y = f ( t ) : y = df dt , y = d 2 f dt 2 , . . . , y ( n ) = d n f dt n etc . ; for u = f ( x, y ) : u x = ∂f ∂x , u y = ∂f ∂y , u xx = 2 f ∂x 2 , etc . Here are some examples of differential equations: y + y = sin( t ) for y = f ( t ) , (1) u xx + u yy = 0 for u = f ( x, y ) , (2) t 2 y + yy = 0 for y = f ( t ) , (3) u t - uu xxxx = x sin t for u = f ( x, t ) . (4) A differential equation may or may not involve f or any particular derivative of f . In the differential equation (2) above, the variable u and the derivative u xy do not appear, while in (3) the derivative y does not appear. But for an equation to be a differential equation, at least some derivative of f must be present. In each of the four examples given above, at least some derivative of f appears.
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