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In this paper, a new approach for solving the second order nonlinear ordinary differential equation y’’ + p(x; y)y’ = G(x; y) is considered. The results obtained by this approach are illustrated by examples and show that this method is powerful for this type of equations.

Exact solutions have always played and still play an important role in properly understanding the qualitative features of many phenomena and processes in various fields of natural science. Exact solutions of nonlinear equations, including those without a clear physical sense which do not correspond to real phenomena and pro- cesses, play an important role of test problems for verifying the correctness and assessment of accuracy of various numerical, asymptotic, and approximate methods. Moreover, the model equations admitting exact solutions serve as the basis for the development of new numerical, asymptotic, and approximate methods, which, in turn, enable us to study more complicated problems having no analytical solutions [

To look for exact solution of (1) the authors introduced the substitution

and have looked for a solution of the Riccati equation

In this paper, we generalize the idea of [

which can be written as

where

In this section, we propose an algorithm that enables us to reduce the Equations (4) and (5) by looking for solutions of the partial differential equations

Theorem 1. If v(x; y) is any solution of (6) where (x; y) is a solution of (7), then Equation (4) can be reduced to a first order equation.

Proof. In order to prove this theorem, consider the transformation

if we differentiate both sides of (8) with respect to x we obtain

substituting (4) and (8) in (9), we have

assuming that

Theorem 2. If

Then (5) can be reduced to a first equation.

Proof. From theorem (1) the associated equation with

which has a solution

Theorem 3. If

Then Equation (5) can be reduced to first order equation.

Proof. Equation (5) can be written as

applying theorem (1), we have that

solving (13) for

Theorem 4. If

Proof. Applying theorem (2) the result follows. ■

In this section, we give some examples on our approach for reduction and finding solutions of nonlinear second order ordinary differential equations, these equations and more equations that can be easily solved by this method can be found in [

Example 1. Consider the equation

comparing with Equation (4) we note that

First, we solve

the associated ratios with Equation (17) are

from which, we find that

Second, we solve

the associated ratios with Equation (19) are

from which, we find that

Finally, we substitute

Example 2. Consider the equation

this equation can be written as

comparing with Equation (5) we have that

The equation associated with

from which we find that

we look for a solution of the form

substituting

Thus,

from which we find that

so,

and two cases are considered,

Example 3. Consider the equation

Equation (36) can be written as

Comparing with Equation (5) we have

which implies that

Differentiating both sides of (39), we have

Assuming that

thus, Equation (36) reduced to the first order exact ordinary differential equation

which has the solution

In this article, a new method is considered for solving second order nonlinear ordinary differential equations. The small size of computation in comparison with the computational size required by other analytical methods [