Explicit solutions for differential equations

Abstract

Advantages exist in use of the decomposition method [1, 2] for solutions of differential equations. Even for the trivial case of solution of first-order separable differential equations the decomposition solutions are more useful because of the resulting convenient computable explicit solutions. The same techniques and benefits apply to the algebraic equations obtained by transform methods in solving differential equations. A comparison is made also between solutions by integrating factor and decomposition, and it is shown that decomposition is an obvious recourse when an integrating factor is not available. To show advantages of the procedure, a differential equation solvable by several methods and involving a logarithmic nonlinearity is solved by Adomian's decomposition for comparisons. The decomposition method will also solve higher-order differential equations and partial differential equations with logarithmic or even composite nonlinearities [2] when the other methods fail.