Page 187 - Foundations Of Differential Calculus
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170    9. On Differential Equations
        and if this is multiplied by y,wehave

                           3      2                2
                          y dy + x ydx = axy dy + ay dx.
                           3                3
        If we substitute for y its value 3axy − x , we obtain the new equation
                                   3      2        2
                          2ax dy − x dy + x ydx = ay dx.
                                                                3
        If we multiply this equation again by y and then substitute for y its value,
        we have
                              3
                        2
                                                          3
                                      2 2
                                                2
                   2axy dy − x ydy + x y dx =3a xy dx − ax dx.
        In general, if P, Q, R represent any functions of x and y, and if the differ-
        ential equation is multiplied by P, then
                           2       2
                         Py dy + Px dx = aPx dy + aPy dx.
                   3
                        3
        Now, since x + y − 3axy = 0, we also have
                            3  3
                          x + y − 3axy (Qdx + Rdy)=0.
        When these equations are added to each other we obtain a general differ-
        ential equation that arises from the given finite equation

                                3
               2
                                        3
             Py dy − aPx dy + Rx dy + Ry dy − 3aRxy dy
                      2               3        3
                 + Px dx − aPy dx + Qx dx + Qy dx − 3aQxy dx =0.
        287. It is possible to find an infinite number of differential equations
        through differentiation from the same finite equation, since before differen-
        tiation the equation can be multiplied or divided by an arbitrary quantity.
        Thus, if P is any function of x and y, so that dP = pdx + qdy, and if
        the finite equation is multiplied by P and then differentiated, we obtain a
        general differential equation that takes on an infinite number of forms in-
        sofar as P takes on one or another function. This multiplicity is increased
        infinitely if this equation is added to the original finite equation multiplied
        by the formula Qdx+Rdy, where Q and R can be any functions of x and y.
        Although in all of these equations the relation between dy and dx remains,
        and this is determined by the differential of the function y in the original
        finite equation, nevertheless, the differential of y could be determined by
        other finite equations. The reason for this is better explained in integral
        calculus.
        288. Not only can we obtain an innumerable number of equations from
        a single finite equation, but we can also find an infinite number of finite
        equations that lead to the same differential equation. For instance, these
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