The World of Mathematical Equations
> Interesting Papers
> N.A. Kudryashov. Seven common errors in finding exact solutions of nonlinear differential equations
> 4. Third error...
4. Third error: some authors omit arbitrary constants after integration of equation
Reductions of nonlinear evolution equations to nonlinear ordinary differential equations can be often integrated. However, some authors assume, that the arbitrary constants of integration are equal to zero. This error potentially leads to the loss of the arbitrary constants in the final expression. So the solution obtained in such way is less general than it could be. The third common error can be formulated as follows.
Third error. Some authors omit the arbitrary constants after integrating of the nonlinear ordinary differential equations.
Example 3a. Reduction of the Burgers equation by Soliman .
Soliman  considered the Burgers equation
to solve this equation by so called ”the modified extended tanh - function method”.
It is well known, that by using the Cole-Hopf transformation [52, 53]
we can write the equality
From the last relation we can see, that each solution of the heat equation
gives the solution of the Burgers equation by formula (4.2).
However to find the solutions of the Burgers equation Soliman  used the travelling wave solutions , and from Eq.(4.1) after integration with respect to the author obtained the equation in the form
The constant of integration he took to be equal to zero. The general solution of Eq.(4.5) takes the form
where C2 is an arbitrary constant.
The general solution of Eq.(4.5) has the only arbitrary constant. But if we take nonzero constant of integration in Eq.(4.5), we can have two arbitrary constants in the solution.
Example 3b. Reduction of the (2+1) - dimensional Konopelchenko - Dubrovsky equation by Abdou 
Using the wave solutions
the author  looked for the exact solutions of Eq.(4.7). After integration with respect to he obtained the second order differential equation
but the zero constant of integration was taken. Abdou  used the Exp - function method to look for solitary wave solutions of Eq.(4.9).
However multiplying Eq.(4.9) on and integrating this equation with respect to again, we have the equation
where C2 is a constant of integration. The general solution of Eq.(4.10) is expressed via the Jacobi elliptic function .
Example 3c. Reduction of the Ito equation by Wazwaz 
The author  looked for the solutions of Eq. (4.11) taking into account the travelling wave
Substituting (4.12) into (4.11) Wazwaz obtained when and the equation in the form
Integrating Eq.(4.13) twice with respect to one can have the equation
where C8 and C9 are arbitrary constants. Denoting in Eq. (4.14) we get the equation
The general solution of this equation was discussed above in example 2b.
However the author  looked for solutions of Eq. (4.14) for C8 = 0 and C9 = 0 taking into consideration the tanh - coth method and did not present the general solution of Eq. (4.13).
Example 3d. Reduction of the Boussinesq equation by Bekir 
Using the wave variable the author  got the equation
To look for the solitary wave solutions of Eq.(4.17) the author  used the - method, but he have been omitted two arbitrary constants after the integration.
In fact, from Eq.(4.16) we obtain the second order differential equation in the form
The general solution of Eq.(4.18) is expressed at via the first Painlevé transcendents. In the case solutions of Eq.(4.18) is determined by the Weierstrass elliptic function. All possible solutions of Eq. (4.16) were obtained in work .
The EqWorld website presents extensive information on solutions to
various classes of ordinary differential equations, partial differential
equations, integral equations, functional equations, and other mathematical
Copyright © 2004-2017 Andrei D. Polyanin