Bernoulli Differential Equations Problems And Solutions Pdf

[Nickie Koskinen]

ABernoulli differential equation is an equation of the form \( y' + a(x)\,y = g(x)\,y^\nu , \) where a(x) are g(x) are given functions, and the constant ν is assumed to be any real number other than 0 or 1. Bernoulli equations have no singular solutions.

Bernoulli returned to Switzerland and began teaching mechanics at the University in Basel from 1683. In 1684 he married Judith Stupanus; and they had two children. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus. Jacob collaborated with his brother on various applications of calculus. However the atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical challenges to test each other's skills. By 1697, the relationship had completely broken down.

Jacob Bernoulli is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. In May 1690, in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. He also discovered the fundamental mathematical constant e, which Euler later denoted by e. Jacob made a great contribution to calculus. By 1689 he had published important work on infinite series and showed that harmonic series diverges, which had actually been proved by Mengoli 40 years earlier. Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced.

Jacob Bernoulli chose a figure of a logarithmic spiral (its equation in polar coordinates is \( r = a\, e^b\,\theta \) ) and the motto Eadem mutata resurgo ("Changed and yet the same, I rise again") for his gravestone; the spiral executed by the stonemasons was, however, an Archimedean spiral, \( r = a + b\, \theta . \)

Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.Moreover, they do not have singular solutions---similar to linear equations. There are two methods known to determine its solutions: one was discovered by himself, and another is credited to Gottfried Leibniz (1646--1716).

Example: Consider the differential equation \( y\, y' = y^2 + e^x . \) To solve it, we first use the Leibniz substitution: \( u = y^2 \quad \Longleftrightarrow \quad y = u^1/2 . \) Then \( y' = \frac12\, u^-1/2 u' \quad \Longrightarrow \quad y\,y' = \frac12\, u' \) and we get the linear differential equation \[ \frac12\, u' = u + e^x . \] To solve it, we first find an integrating factor by solving the corresponding separable equation \[ \frac12\, \mu' = -\mu \qquad \Longrightarrow \qquad \frac\text d \mu\mu = -2\,\text dx \qquad \Longrightarrow \qquad \mu (x) = e^-2x . \] Multiplying both sides of the equation \( u' - 2\,u = 2\, e^x \) by the integrating factor, we obtain an exact equation \[ \frac\text d\text dx\left( e^-2x u(x) \right) = 2\, e^-x \qquad \Longrightarrow \qquad e^-2x u(x) = C -2\, e^-x \qquad \Longrightarrow \qquad u (x) = C\,e^2x - 2\, e^x , \] where C is a constant of integration. Returning to the original function, we find the general solution to the given differential equation \( y = \pm \sqrtC\,e^2x - 2\, e^x . \) Now we demonstrate the application of the Bernoulli method by seeking the solution as the product y = u v, where u is a solution of the "linear part:"

Example: The four streamlines (corresponding to the values of an arbitrary constant C=1,2,3,4) from the general solution y=1/(x Sqrt[C-2 ln[x]]) of the Bernoulli equation

x y' =x^2 y^3 -y can be plotted with one command:

Plot[-1/(x Sqrt[#1 - 2 Log[x]] &) /@ 1, 2, 3, 4, x, .3, 10 ] Example: Bernoulli equation with a=4. Example: Consider the Bernoulli equation \( y' +x\,y=x\, y^4 . \) Using Leibniz substitution \[ u = y^-3 \qquad \Longleftrightarrow \qquad y = u^-1/3 \qquad \Longrightarrow \qquad y' = \frac-13\,u^-4/3 u' , \] we reduce the given differential equation to a linear one.

The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Bcklund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Bcklund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics.

Many well-known NLPDEs can be handled by those traditional methods. However, there is no unified method which can be used to deal with all types of NLPDEs. Moreover, we always encounter the fractional NLPDEs, the NLPDEs which have nonlinear terms of any order or peaked wave solutions. It is significant to construct traveling wave solutions of NLPDEs by a uniform method. Based on those problems, the Riccati-Bernoulli sub-ODE method is firstly presented.

In this paper, the Riccati-Bernoulli sub-ODE method is proposed to construct traveling wave solutions, solitary wave solutions, and peaked wave solutions of NLPDEs. By using a traveling wave transformation and the Riccati-Bernoulli equation, NLPDEs can be converted into a set of algebraic equations. Exact solutions of NLPDEs can be obtained by solving the set of algebraic equations. The Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers (ZK-Burgers) equation are chosen to illustrate the validity of the Riccati-Bernoulli sub-ODE method. A Bcklund transformation of the Riccati-Bernoulli equation is given. If we get a solution of NLPDEs, we can search for a new infinite sequence of solutions of the NLPDEs by using a Bcklund transformation.

The remainder of this paper is organized as follows: the Riccati-Bernoulli sub-ODE method is described in Section 2. In Section 3, a Bcklund transformation of the Riccati-Bernoulli equation is given. In Sections 4-7, we apply the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized ZK-Burgers equation, respectively. In Section 8, our results are compared with the first integral method, the \(( \fracG'G )\)-expansion method, and physical explanations of the obtained solutions are discussed. In Section 9, some conclusions and directions for future work are given.

The Eckhaus equation was found [26] as an asymptotic multiscale reduction of certain classes of nonlinear Schrdinger type equations. A lot of the properties of the Eckhaus equation were obtained [27]. The Eckhaus equation can be linearized by making some transformations of dependent variables [28]. An exact traveling wave solution of the Eckhaus equation was obtained by the \(( \fracG'G )\)-expansion method [8] and the first integral method [5].

The generalized Ostrovsky equation is a model for the weakly nonlinear surface and internal waves in a rotating ocean. Exact peaked wave solutions were obtained by the undetermined coefficient method [32].

The generalized ZK-Burgers equation retains the strong nonlinear aspects of the governing equation in many practical transport problems such as nonlinear waves in a medium with low-frequency pumping or absorption, transport and dispersion of pollutants in rivers, and sediment transport. Wang et al. obtained a solitary wave of the generalized ZK-Burgers equation with a positive fractional power term by using the HB method and with the aid of sub-ODEs [33].

In this section, the physical interpretation of the results of Sections 4-7 are given, respectively. We will compare the Riccati-Bernoulli sub-ODE method with the \(( \fracG'G )\)-expansion method, the first integral method, and so on. Some of our obtained exact solutions are in the figures represented with the aid of Maple software.

Moreover, by using a Bcklund transformation, we can get an infinite sequence of solutions of these NLPDEs which cannot be obtained by the \(( \fracG'G )\)-expansion method and the first integral method. The graphical demonstrations of some obtained solutions are shown in Figures 1-4.

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