Dear Dasre, If your ODEs are initial ODEs it is easy to use Adomian Decomposition Method (ADM), but if you want to solve a boundary value ODE you need to further analysis. The linearity of your ODEs is also important.

There are several types of ADM, which one of the well known ADM methods is modified ADM which proposed by WazWaz. Some times the ADM polynomials also fails to obtain a solution for boundary value problems and some times they also fails to obtain a solution of an initial ODE on the entire domain of analysis.

I suggest you to expose your ODEs and related boundary conditions, then may be we could help you to solve your problem.

There are many papers on the application of ADM to solve ODE.

you can get them

You may also be interested in some of my other publications:

A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011

R. Rach

Kybernetes, Volume 41, Issues 7/8, 2012, Pages 1087-1148

Select Supplemental Bibliography of the Adomian Decomposition Method - Post-2011.pdf

A new modified Adomian decomposition method for higher-order nonlinear dynamical systems

J.-S. Duan, R. Rach, A.-M. Wazwaz

CMES: Computer Modeling in Engineering & Sciences, Vol. 94, No. 1, 2013, Pp. 77-118.

24. Solution of higher-order, multipoint, nonlinear boundary value problems with high-order Robin-type boundary conditions by the Adomian decomposition method

R. Rach, J.-S. Duan, A.-M. Wazwaz

Applied Mathematics and Information Sciences, Vol. 10, No. 4, 2016, Pp. 1231-1242.

23. Higher-order numeric solutions of the Lane-Emden type equations derived from the multi-stage modified decomposition method

J.-S. Duan, R. Rach, A.-M. Wazwaz

International Journal of Computer Mathematics, International Journal of Computer Mathematics, Volume 94, No. 1, 2017, 197-215.

22. The degenerate form of the Adomian polynomials in the power series method for nonlinear ordinary differential equations

J.-S. Duan, R. Rach

Journal of Mathematics and System Science, Volume 5, Pages 411-428, 2015.

21. On the solution of non-isothermal reaction-diffusion model equations in a spherical catalyst by the modified Adomian method

R. Rach, J.-S. Duan, A.-M. Wazwaz

Chemical Engineering Communications, Volume 202, Issue 8, 2015, Pages 1081-1088.

20. A new parametric algorithm for isothermal flash calculations by the Adomian decomposition of Michaelis-Menten type nonlinearities

H. Fatoorehchi, H. Abolghasemi, R. Rach

Fluid Phase Equilibria, Vol. 395, 2015, Pp. 44-50.

19. Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method

J.-S. Duan, R. Rach, A.-M. Wazwaz

Journal of Mathematical Chemistry, Vol. 53, 2015, Pages 1054-1067.

18. A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method,

J.-S. Duan, R. Rach, A.-M. Wazwaz,

Open Engineering, Volume 5, Issue 1, 59-74, 2014, DOI: 10.1515/eng-2015-0007.

17. The Volterra integral form of the Lane-Emden equation: New derivations and solution by the Adomian decomposition method

R. Rach, A.M. Wazwaz and J.S. Duan

Journal of Applied Mathematics and Computing, Volume 47, Issue 1, 2015, Pages 365-379.

16. A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

R. Rach, A.-M. Wazwaz, J.-S. Duan

International Journal of Biomathematics, Vol. 07, No. 02, March 2014, 1450020 (2014) [12 pages].

15. Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method

R. Rach, J.-S. Duan, A.-M. Wazwaz

Journal of Mathematical Chemistry, Volume 52, Issue 1, 2014, Pages 255-267.

14. A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method

A.-M. Wazwaz, R. Rach, J.-S. Duan

Mathematical Methods in the Applied Sciences, Vol. 37, Issue 1, 2014, Pages 10-19.

13. A new modified Adomian decomposition method for higher-order nonlinear dynamical systems

J.-S. Duan, R. Rach, A.-M. Wazwaz

CMES: Computer Modeling in Engineering & Sciences, Vol. 94, No. 1, 2013, Pp. 77-118.

12. A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions

J.-S. Duan, R. Rach, A.-M. Wazwaz, T. Chaolu, Z. Wang

Applied Mathematical Modelling, Vol. 37, Nos. 20/21, 2013, Pages 8687-8708.

11. The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations

J.-S. Duan, T. Chaolu, R. Rach, L. Lu

Computers and Mathematics with Applications, Volume 66, Issue 5, 2013, pp. 728-736.

10. On the effective region of convergence of the decomposition series solution

J.-S. Duan, R. Rach, Z. Wang

Journal of Algorithms and Computational Technology, Volume 7, 2013, Pages 227-247.

9. A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011

R. Rach

Kybernetes, Volume 41, Issues 7/8, 2012, Pages 1087-1148.

8. Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions

A.-M. Wazwaz, R. Rach, J.-S. Duan

Applied Mathematics and Computation, Volume 219, Issue 10, 2013 Pages 5004-5019.

7. A review of the Adomian decomposition method and its applications to fractional differential equations

J.-S. Duan, R. Rach, D. Baleanu, A.-M. Wazwaz

Communications in Fractional Calculus, Volume 3, Issue 2, 2012, Pages 73-99.

6. A reliable modification of the Adomian decomposition method for higher-order nonlinear differential equations

R. Rach, A.-M. Wazwaz, J.-S. Duan

Kybernetes, Volume 42, Issues 1/2, 2013, Pages 282-308.

5. Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems

J.-S. Duan, R. Rach, A.-M. Wazwaz

International Journal of Non-Linear Mechanics, Volume 49, 2013, Pages 159-169.

4. Higher-order numeric Wazwaz-El-Sayed modified Adomian decomposition algorithms

J.-S. Duan, R. Rach

Computers and Mathematics with Applications, Volume 63, Issue 11, 2012, Pages 1557-1568.

3. Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method

J.-S. Duan, T. Chaolu, R. Rach

Applied Mathematics and Computation, Volume 218, Issue 17, 2012, Pages 8370-8392.

2. A new modification of the Adomian decomposition method for solving boundary value problems for higher order differential equations

J.-S. Duan, R. Rach

Applied Mathematics and Computation, Volume 218, Issue 8, 2011, Pages 4090-4118.

1. New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

J.-S. Duan, R. Rach

Applied Mathematics and Computation, Volume 218, Issue 6, 2011, Pages 2810-2828.

See also papers by Jun-Sheng Duan on automated generation of the Adomian polynomials!

See the attached files. I am sure that they will be enough. Also using modified version depends on your problem. Some times MADM not better than ADM. You should check them.

For solving linear and nonlinear ODEs subject to boundary conditions, I recommend that you use the Duan-Rach modified Adomian decomposition method as discussed in:

1. J.-S. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order differential equations, Applied Mathematics and Computation, Vol. 218, No. 8, 2011, Pp. 4090-4118.

https://doi.org/10.1016/j.amc.2011.09.037

2. J.-S. Duan, R. Rach, A.-M. Wazwaz, Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems, International Journal of Non-Linear Mechanics, Vol. 49, 2013, Pp. 159-169.

https://doi.org/10.1016/j.ijnonlinmec.2012.10.003

3. J.-S. Duan, R. Rach, A.-M. Wazwaz, Temuer Chaolu, Z. Wang, A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions, Applied Mathematical Modelling, Vol. 37, Nos. 20/21, 2013, Pp. 8687-8708.

https://doi.org/10.1016/j.apm.2013.02.002

4. A.-M. Wazwaz, R. Rach, J.-S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Applied Mathematics and Computation, Volume 219, Issue 10, 2013 Pp. 5004-5019.

https://doi.org/10.1016/j.amc.2012.11.012

5. Jun-Sheng Duan, Zhong Wang, Shou-Zhong Fu, Temuer Chaolu, Parametrized temperature distribution and efficiency of convective straight fins with temperature-dependent thermal conductivity by a new modified decomposition method, International Journal of Heat and Mass Transfer, Vol. 59, April 2013, Pp. 137-143.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11.080

6. J.-S. Duan, R. Rach, A.-M. Wazwaz, A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, Vol. 5, No. 1, 2014, Pp. 59-74.

https://doi.org/10.1515/eng-2015-0007

7. R. Rach, J.-S. Duan, A.-M. Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method, Journal of Mathematical Chemistry, Volume 52, No.1, 2014, Pp. 255-267.

https://doi.org/10.1007/s10910-013-0260-6

8. R. Rach, A.-M. Wazwaz, J.-S. Duan, A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, International Journal of Biomathematics, Vol. 07, No. 02, March 2014, 1450020 (2014), [12 pages].

https://doi.org/10.1142/S179352451450020X

9. J.-S. Duan, R. Rach, A.-M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method, Journal of Mathematical Chemistry, Vol. 53, No. 4, 2015, Pp. 1054-1067.

https://doi.org/10.1007/s10910-014-0469-z

10. R. Rach, J.-S. Duan, A.-M. Wazwaz, On the solution of non-isothermal reaction-diffusion model equations in a spherical catalyst by the modified Adomian method, Chemical Engineering Communications, Vol. 202, No. 8, 2015, Pp. 1081-1088.

https://doi.org/10.1080/00986445.2014.900054

11. R. Rach, J.-S. Duan, A.-M. Wazwaz, Solution of higher-order, multipoint, nonlinear boundary value problems with high-order Robin-type boundary conditions by the Adomian decomposition method, Applied Mathematics and Information Sciences, Vol. 10, No. 4, 2016, Pp. 1231-1242.

https://doi:10.18576/amis/100403

12. Simulation of large deflections of a flexible cantilever beam fabricated from functionally graded materials by the Adomian decomposition method

R. Rach, J.-S. Duan, A.-M. Wazwaz

International Journal of Dynamical Systems and Differential Equations, Article In Press, 2019,

https://www.inderscience.com/info/ingeneral/forthcoming.php?jcode=ijdsde