Advances in Pure Mathematics, 2011, 1, 3-4
doi:10.4236/apm.2011.11002 Published Online January 2011 (http://www.scirp.org/journal/apm)
New Solutions to Nonlinear Ordinary
Differential Equations
Moawia Alghalith
Economics Department, UWI, Trinidad, Cuba
Received January 11, 2011; revised January 19, 2011; accepted January 29, 2011
Abstract
In contrast to the Euler method and the subsequent methods, we provide solutions to nonlinear ordinary dif-
ferential equations. Consequently, our method does not require convergence. We apply our method to a
second-order nonlinear ordinary differential equation ODE. However, the method is applicable to higher or-
der ODEs.
Keywords: Ordinary Differential Equations ODE; Euler ’s Method
1. Introduction
There are several methods of solving nonlinear ordinary
differential equations, such as the Euler method, Runge-
Kutta methods and linear multistep methods. For a de-
tailed description of these methods, see, for example,
Kaw and Kalu (2009), Cellier and Kofman (2008) and
Butcher (2008). However, these methods are approxima-
tions of the so lution and t h us the y require the ass umptio n
of convergence to the solution. Consequently, numerical
methods, based on real data, are needed to obtain a solu-
tion.
In this paper, in contrast to the previous methods, we
present solutions to nonlinear ordinary differential equa-
tions without the requirement of convergence and with-
out the need to numerical methods. In addition, our me-
thod is far s imp ler t han the e xisti ng methods.
2. The Model
We attempt to solve the following nonl inear or dinary dif-
ferential equation (higher-order equations can also be
used)
( )( )( ) ( )
, ,,,ysf sysyssS
′′ ′
= ∈
Consider the following Taylor expansion of y around
α
()( )( )()( )()()
2
1,
2
ys yysysRs α αααα
′ ′′ =+−+− +
where
( )( )()()()
( )()
2
1
2
Rsys yys
ys
α αα
αα
′ =−+ −
′′ +−
is the remainder. Our intermediate goal is to minimize
the remainder (in abs olute value) with respect to ti me s
( )
min
s
Rs
The first-order condition yields
( )( )
( )( )
( )
0,Rsys yys ααα
∗∗ ∗
′′′ ′′
=− −−=
and thus
( )
( )( )
( )
( )
( )
2
1.
2
ys yysys
ααα αα
∗ ∗∗
′ ′′ = +−+− (1)
Since
( )( )( ) ( )
,,,yfy y ααα α
′′ ′
=
we obtain
( )
( )( )
( )
( )( ) ( )
( )
2
1
,,.
2
ys yys
fy ys
ααα
αα αα
∗∗
∗
′
=+−
′
+−
But
( )( ) ( )
,y gy α αα
′=
(this is a result of integrating
(1)) and thus subst itut in g this into (1), we obtain
( )
( )( )
( )
( )
( )( ) ( )
( )
2
,
1
,,.
2
ysygys
fy ys
α ααα
αα αα
∗∗
∗
=+−
′
+−
The initial values
( )( ) ( )
, ,,,y gy αα αα
and
( )( ) ( )
,,fy y
αα α
′
are known (assumed by the pre-
vious literature). Thus, this is a solution to (1). The ex-
tension of this method to higher-order differential equa-
tions is straightforward.

M. ALGHALITH
4
3. Practical Examples
For simplicity, we present two first-order numerical ex-
amples. Using the above procedure, the solution for a
first-order differential equation takes the form
( )
( )( ) ( )
( )
,,ysyg ys α ααα
∗∗
=+−
(2)
and the minimization necessary condition
( )
( ) ( )
, 0.ysgy αα
∗
′−=
Example 1.
( )
2
ys s
′=
with initial values
1, α =
( )
1y α =
and
( ) ( )
, 1.gy αα =
It is well established in the literature that the solution of
the differential equation depends on the initial values,
and that different initial values produce different solu-
tions. Therefore, from the necessary condition,
*2
1s=
and thu s
*
1s= −
, since
*
s α ≠
by construction. Hence,
using (2), we obtain
( )
( )
1 11 11.ys
∗
= +⋅−−=−
Example 2.
( )
1yss s
′=−+
with in i tial values
0, α =
( )
0y α =
and
( ) ( )
, 1.gy αα =
Hence,
**
11ss− +=
; therefore , *1s= and thus
( )
( )
0110 1.ys
∗
= +⋅−=
4. Referen ces
[1] J. C. Butcher, “Numerical methods for ordinary differen-
tial equations,” Wiley, W. Sussex, England, 2008.
[2] F. Cellier and E. Kofman, “Continuous system simula-
tion,” Springer Verlag, New York, NY, 2006.
[3] A. Kaw and E. Kalu, “Numerical methods with applica-
tions,” www.autarkaw.com, 2009.