Exact solutions of sixth and fifth degree equations

The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

On the Unsolvable Quintic Equation x^5 + x^2 + x + 1/π = 0

According to Galois’ theory, a solution x for the quintic equation cannot be solved for in radicals. Here we do not solve for x. We construct it. We note that radical numbers a and b can be found, such that a line that connects the points (a; f(a)) and (b; f(b)) passes through the interior point (x; 0), a root of the quintic function f(x). We then test this on f(x) = x5 +x2 + x+1/π. The values found for a and b, converted to decimal form, are a = −0.6012227544458956 and b = −0.6012348965947528, from which it is determined that x = −0.6012235335340363. The corresponding value, obtained through the software Mathematica numerical routines, is xn = −0.6012235335340362

The Lambert function, the quintic equation and the proactive discovery of the Implicit Function Theorem

One of the problems on which a great deal of focus is being placed today, is how to teach Calculus in the presence of the massive diffusion of Computer Algebra tools and online resources among students. The essence of the problem lies in the fact that, during the problem solving activities, almost all undergraduates can be exposed to certain ''new'' functions, not typically treated at their level. This, without being prepared to handle them or, in some cases, even knowing the meaning of the answer provided by the computer system used. One of these functions is Lambert’s \(W\) function, undoubtedly due to the elementary nature of its definition. In this article we introduce \(W\), in a way that is easy to grasp for first year undergraduate students and we provide some general results concerning polynomial-exponential and polynomial-logarithmic equations. Among the many possible examples of its applications, we will see how \(W\) comes into play in epidemiology in the SIR model. In the second part, using more advanced concepts, we motivate the importance of the Implicit Function Theorem, using it to obtain the power series expansion of the Lambert function around the origin. Based on this approach, we therefore also provide a way to obtain the power series expansion of the inverse of a given smooth function \(f(y)\), when it is assumed that \(f(0)=0,\,f'(0)\neq0\), aided by the computational power of Mathematica\(_\circledR\). Basically, in this way, we present an alternative approach to the Lagrange Bürman Inversion Theorem, although in a particular but relevant case, since the general approach is not at an undergraduate level. A number of good references are [<a href="#1">1</a>, pp. 23-28] and [<a href="#2">2</a>], where the Lambert function is applied. Finally, these skills are used to take into consideration the particular quintic equation in the unknown \(y\) presented by F. Beukers [<a href="#3">3</a>]. Namely, we consider \(x(1+y)^5-y=0\) as an example of an equation for which the power series representation of one of its real solutions is known, calculating, with the same method used for the Lambert function, the first terms of its power series representation.

A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

Insert a Root to Extract a Root of Quintic Quickly

The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u2− g2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed

The current paper investigates the dynamics of the dissipative system with a cubic nonlinear time-delayed of the type of the damping Duffing equation. A coupling between the method of the multiple scales and the homotopy perturbation has been utilized to study the complicated dynamic problem. Through this approach, a cubic nonlinear amplitude equation resulted in at the first-order of perturbation; meanwhile, a quintic equation appears at the second-order of perturbation. These equations are combined into one nonlinear quintic Landau equation. The polar form solution is used, and linearized stability configuration is applied to the nonlinear amplitude equation. Also, a second-order approximate solution is achieved. The numerical illustrations showed that the damping, delay coefficient, and time delay play dual roles in the stability behavior. In addition, the nonlinear coefficient plays a destabilizing influence.