Seven common errors in finding exact solutions of nonlinear differential equations
Let us assume that Eq. (5.1) is autonomous and this equation admits the shift of the independent variable (where C2 is an arbitrary constant). This means that the constant C2 added to the variable in Eq.(5.1) does not change the form of this equation. In this case Eq.(5.1) is invariant under the shift of the independent variable.
Taking this property into account, we obtain the advantage for the solution of Eq. (5.1). If we know a solution of Eq. (5.1), then for the autonomous equation we have a solution of this equation with additional arbitrary constant .
The main feature of the autonomous equation is that the fact that the solution is more general, then .
The error discussed often leads to a huge amount of different expressions for the solutions of nonlinear differential equations instead of choosing one solution with an arbitrary constant .
The application of the tanh - function method [28, 29, 30] for finding the exact solutions allows us to have the special solutions of nonlinear differential equations as a sum of hyperbolic tangents . However for the autonomous equation such types of the solutions can be taken as the more general solution in the form .
Example 4a. Solution of the Riccati equation
In fact, the general solution of Eq. (5.2) takes the form
Example 4b. Solution of the Cahn - Hilliard equation by Ugurlu and Kaya 
the authors obtained the exact solutions of the equation
They found eight solutions of Eq.(5.11) at c = 1. Six solutions are the following
However all these solutions can be written as the only solution with an arbitrary constant z0
Note, that Eq.(5.11) at c = 1 takes the form
Twice integrating Eq.(5.16) with respect to z, we have
where C1 and C2 are arbitrary constants. At C1 = 0 the solution of Eq.(5.17) is expressed via the Jacobi elliptic function.
Example 4c. Solution of the KdV - Burgers equation by Soliman 
All these solutions can be written in the form
We can see, that these solutions do not differ, if we take the constant into account in one of these solutions.
Example 4d. Solution of the combined KdV - mKdV equation by Bekir 
All these solutions can be written as the only solution with an arbitrary constant
Example 4e. Solution of the coupled Hirota — Satsuma — KdV equation by Bekir 
(where k, and are arbitrary constants)
Example 4f. ”Twenty seven solution” of the ”generalized Riccati equation” by Xie, Zhang and Lü .
The authors  ”firstly extend” the Riccati equation to the ”general form”
where r, p and q are the parameters. They ”fortunately find twenty seven solutions” of Eq.(5.36).
It was very surprised that the authors are not aware that the solution of Eq.(5.36) was known more then one century ago. It is very strange but these 27 solutions was repeated by Zhang  as the important advantage.
Let us present the general solution of Eq.(5.36). Substituting
into Eq.(5.36) we have
in Eq.(5.38) we obtain the linear equation of the second order
The general solution of Eq.(5.40) is well known
Example 4g. ”New solutions and kinks solutions” of the Sharma — Tasso — Olver equation by Wazwaz .
Using the extended tanh method Wazwaz  have found 18 solitary wave and kink solutons of the Sharma — Tasso —- Olver equation
Taking the travelling wave solution , into account the author considered the nonlinear ordinary differential equation in the form
However Eq.(5.44) can be transformed to the second - order linear differential equation (see, example 2f)
by the transformation
We can see, that there is no need to write a list of all possible expressions for the solutions at the given . It is enough to present the solution of the equation with an arbitrary constant. Moreover, the solution with arbitrary constants looks better.
The simple and powerful tool to remove this error is to plot the graphs of the expressions obtained. The expressions having the same graphs usually are equivalent.
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