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# S12 - 1.2 n-Parameter Family of Solutions General Solution...

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

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