The particular solution in the transformed variable is then w p ( t ) = 144 t 4 . So one assumes a solution of the form w = w p ( 1 + α α t λ λ ) and if this is substituted in the above equation and the coefficients of α α are equated, one obtains the value for λ λ , which is given by the roots of the equation λ λ 2 + 7 λ λ − − 6 = 0 . The two roots are λ λ 1 = 0.772 , λ λ 2 = − − 7.772 , where we need to take the positive root to avoid the singularity at the origin. This solution already satisfies the first boundary condition ( w ( 0 ) = 0 ), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary n