Consider the initial value problem of first order ordinary differential equation in the form
$${y}^{{\prime }}+{q}^{{\prime }}\left(x\right)y=g\left(x\right)+f\left(x,y\right) \left(1\right)$$
Where a is given constant and \(g\left(x\right), f\left(x,y\right)\) is a real function.
We rewrite (1) in the form
$$Ly=g\left(x\right)+f\left(x,y\right) \left(2\right)$$
Where the differential operator L is defined by
$$L\left(.\right)={e}^{-q\left(x\right)}\frac{d}{dx}{e}^{q\left(x\right)}\left(.\right) \left(3\right)$$
The inverse operator \({\text{L}}^{-1}\) is therefore considered integral operator, as below
$${L}^{-1}\left(.\right)={e}^{-q\left(x\right)}{\int }_{0}^{x}{e}^{q\left(x\right)}(.)dx \left(4\right)$$
Applying \({L}^{-1}\) on (2) we find
$${L}^{-1}\left(Ly\right)={L}^{-1}( g\left(x\right)+f\left(x,y\right) )$$
$$y\left(x\right)=\varPhi \left(x\right)+{L}^{-1}g\left(x\right)+{L}^{-1}f\left(x,y\right) \left(5\right)$$
Such that\(L\left(\varPhi \left(x\right)\right)=0\)
The Adomian decomposition strategy introduces the solution y(x) by an infinite series of components
$$y\left(x\right)=\sum _{n=0}^{\infty }{y}_{n}\left(x\right) \left(6\right)$$
and
$$f\left(x,y\right)=\sum _{n=0}^{\infty }{A}_{n} \left(7\right)$$
where the components \({y}_{n}\left(x\right)\) of the solution will be determined recurrently. Specific algorithms formulate called Adomian polynomials. The following algorithm:
$${ A}_{0}=f\left({u}_{0}\right)$$
$${A}_{1}={f}^{{\prime }}\left({u}_{0}\right){u}_{1}$$
$${A}_{2}={f}^{{\prime }}\left({u}_{0}\right){u}_{2}+\frac{1}{2}{f}^{{\prime }{\prime }}\left({u}_{0}\right){u}_{1}^{2}$$
$${A}_{3}={f}^{{\prime }}\left({u}_{0}\right){u}_{3}+{f}^{{\prime }}\left({u}_{0}\right){u}_{1}{u}_{2}+\frac{1}{3!}{f}^{{\prime }{\prime }{\prime }}\left({u}_{0}\right){u}_{1}^{3}, \left(8\right)$$
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAIAAAATCAMAAACN4iwoAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAbUExURQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAJRR4iAAAAAJdFJOUwAIEHCHn8fP58x0sPQAAAAJcEhZcwAAFxEAABcRAcom8z8AAAAbSURBVBhXY2BhZWDnYGBkYgACZhYGNuLYjEwADmsAfBS7hIoAAAAASUVORK5CYII=)
can be used to construct Adomian polynomials, when \(f\left(u\right)\)is a nonlinear function.
By substituting (6) and (7) into (5),
$$\sum _{n=0}^{\infty }{y}_{n}\left(x\right)=\varPhi \left(x\right)+{L}^{-1}g\left(x\right)+{L}^{-1}\sum _{n=0}^{\infty }{A}_{n} \left(9\right)$$
Through using Adomian decomposition strategy, the components \({y}_{n}\left(x\right)\) can be determined as
$${y}_{0}=\varPhi \left(x\right)+{L}^{-1}g\left(x\right)$$
$${y}_{n+1}\left(x\right)={L}^{-1}{A}_{n} n \ge 0, \left(10\right)$$
Which gives
$${y}_{0}=\varPhi \left(x\right)+{L}^{-1}g\left(x\right),$$
$${y}_{1}={-L}^{-1}{A}_{0} ,$$
$${y}_{2}={-L}^{-1}{A}_{1} ,$$
$${y}_{3}={-L}^{-1}{A}_{2} , \left(11\right)$$
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAIAAAATCAMAAACN4iwoAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAbUExURQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAJRR4iAAAAAJdFJOUwAIEHCHn8fP58x0sPQAAAAJcEhZcwAAFxEAABcRAcom8z8AAAAbSURBVBhXY2BhZWDnYGBkYgACZhYGNuLYjEwADmsAfBS7hIoAAAAASUVORK5CYII=)
From (8) and (11), we can determine the components \({y}_{n}\left(x\right)\), and hence the series solution of \(y\left(x\right)\) in (6) can be immediately obtained.
For numerical purposes, the n-term approximate
$${\varPsi }_{n}=\sum _{n=0}^{n-1}{y}_{n}\left(x\right) \left(12\right)$$
can be used to approximate the exact solution.