S12 - 1.2 n-Parameter Family of Solutions; General...

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1.2 n -Parameter Family of Solutions; General Solution; Particular Solu- tion Introduction. You know from your experience in previous mathematics courses that the calculus of functions of several variables (limits, graphing, differentiation, integration and applications) is more complicated than the calculus of functions of a single variable. By extension, therefore, you would expect that the study of partial differential equations would be more complicated than the study of ordinary differential equations. This is indeed the case! Since the intent of this material is to introduce some of the basic theory and methods for differential equations, we shall conFne ourselves to ordinary differential equations from this point forward. Hereafter the term differential equation shall be interpreted to mean ordinary differential equation. Partial differential equations are studied in subsequent courses. ± We begin by considering the simple Frst-order differential equation y ± = f ( x ) where f is some given function. In this case we can Fnd y simply by integrating: y = ± f ( x ) dx = F ( x )+ C where F is an antiderivative of f and C is an arbitrary constant. Not only did we Fnd a solution of the differential equation, we found a whole family of solutions each member of which is determined by assigning a speciFc value to the constant C . In this context, the arbitrary constant is called a parameter and the family of solutions is called a one-parameter family . Remark In calculus you learned that not only is each member of the family y = F ( x C a solution of the differential equation but this family actually represents the set of all solutions of the equation; that is, there are no other solutions outside of this family. ± Example 1. The differential equation y ± =3 x 2 - sin 2 x has the one-parameter family of solutions y ( x )= ± ( 3 x 2 - sin 2 x ) dx = x 3 + 1 2 cos 2 x + C As noted above, this family of solutions represents the set of all solutions of the equation. ± In a similar manner, if we are given a second order equation of the form y ±± = f ( x ) then we can Fnd y by integrating twice, with each integration step producing an arbitrary constant of integration.
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This note was uploaded on 06/20/2008 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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S12 - 1.2 n-Parameter Family of Solutions; General...

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