Power series methods of solving ordinary differential equations (2) – Leibniz-Maclaurin method

Leibniz-Maclaurin method

Differentiate the given equation n times, using the Leibniz theorem of equation (1), page 400

rearrange the result to obtain the recurrence relation at x = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq5011.tif"/> ,

determine the values of the derivatives at x = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq5012.tif"/> , i.e. find (y)₀ and (y’)₀,

substitute in the Maclaurin expansion for y = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq5013.tif"/> (see page 84),

simplify the result where possible and apply boundary condition (if given).