By W. Richard Kolk

Engineers, scientists, and utilized mathematicians are habitually keen on habit of actual platforms. normally they're going to version the process after which examine the version, hoping to show the system's dynamic secrets and techniques. routinely, linear tools were the norm and nonlinear results have been in simple terms extra peripherally. This bias for linear thoughts arises from the consum mate attractiveness and order in linear subs paces and the splendor of linear indepen dence is simply too compelling to be denied. And the unfairness has been, some time past, for tified by means of the shortage of nonlinear approaches, rendering the learn of nonlinear dynamics untidy. yet now a brand new reputation is being conferred on that non descript patchwork, and the advantage of the hidden surprises is gaining deserved appreciate. With a large choice of person concepts to be had, the coed and the engineer in addition to the scientist and researcher, are confronted with a nearly overwhelming job of which to exploit to assist in achieving an realizing adequate to arrive a delightful outcome. If linear research predicts procedure habit suffi ciently with regards to truth, that's pleasant. within the much more likely case the place nonlin ear research is needed, we think this article fills a massive void. we have now attempted to collect and produce a few order to a large number of info and methods, that even if renowned, is scattered. we've additionally prolonged this information base with new fabric now not formerly published.

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The solution of Chrystal's equation has many fonns depending on the magnitude of the constants A, B, C (ref. 7). For example, if A, B, C satisfy the relation A2 + AB - 4C = 0 a solution of (42) is the parabola 4Bu = _ABx2 ± (d + Bxi (50) where d is a constant. Then the solution of (46), the general nonlinear equation, is given as y = F(u), where u(x) is given by (50). 2 Theorem 4 Using Abel's Equation as the base equation, (u + s(x)]u' + p(x) + q(x)u + r(x)u2 = 0 (51) the general nonlinear equation can be written as: y'{G(y) + s(x)] + Z(y){p(x) + q(x)G(y) + r(x)G(yi] = 0 which has the solution y in terms of the variable transformation y if and only if G, Z satisfy the connective condition Z(y) d/dy (G(y)] = 1 (52) = F(u) (53) ANALYTIC SOLUTIONS TO NONLINEAR DIFFERENTIAL EQUATIONS 39 The functional relation can be written as either = G(F) u =0 or dF/du - Z(F) Proof If Abel's equation (51) is the nonlinear base equation, and y = F(u) is the variable transfonnation, by differentiating and substituting for u, with the definition Fu = Z(y), which corresponds to u = P[dtIZ(t)], there results [r l + sex) y' = -p(x)Z - q(x)Z [dtIZ(t)] r lr [dtIZ(t)] - r(x)Z [dtIZ(t)] r By redefining tenns, the general nonlinear equation having Abel's equation as its base equation is [G(y) + + s(x)]y' Z(y)[p(x) + q(x)G(y) + r(x)G 2 (y)] =0 where G(y), Z(y) must satisfy the connective condition Z(y) dldy [G(y)] = 1 The solution is given as y = F(u), where F is found using u = G(F) or dFldu - Z(F) = 0, which completes the proof.

ANALYTIC SOLUTIONS TO NONLINEAR DIFFERENTIAL EQUATIONS 53 By combining (70) and (71), the following equation results (Hy'YY' + Hy'x - Hy)/Hy'y' Suppose that H = H(x, = p(x)y' + q(x)G(y) + r(x)Z(y) - CZ/Z)y,2 (72) y, y') is separable H = H I(x)H2(y)H3(y') substituting into (72) yields the equation p(x)y' + q(x)G(y) + r(x)Z(y) - (2/Z)y,2 = ~ dH2 (Y' dH3 _ H3) j d2H3 dy' dy,2 H2 dy + (~dHI dH3) jd2H3 HI dx dy' (73) dy,2 To define HI' H 2, and H3, we now compare terms, establish the associated differential equations, and solve.

5 43 DERIVATIVE AND INTEGRAL FUNCTIONAL RELATIONS We have developed solutions of classes of nonlinear ordinary differential equations using both linear and nonlinear ordinary differential equations as base equations. The solutions of the two classes were related to one another by means of a functional relation of the form y = F(u), provided the connective conditions were satisfied. It is logical to consider investigating the use of derivative and integral forms of F to accomplish the same end. 1 Integral Functional Relations Suppose one wishes to investigate the relationship between first- or secondorder base equations and their corresponding solvable nonlinear counterparts.