Solving Equations Involving Power Series

Solving equations involving power series.

Here is a power series:

y=1+Sum[a[i] x^i,{i,3}]+O[x]^4

This gives an equation involving the power series:

D[y,x]^2-y==x

(RefLink[LogicalExpand, paclet:ref/LogicalExpand]) generates a sequence of equations for each power of x:

LogicalExpand[%]

This solves the equations for the coefficients a[i]. You can also feed equations involving power series directly to (RefLink[Solve, paclet:ref/Solve]):

Solve[%]

Some equations involving power series can also be solved using the (RefLink[InverseSeries, paclet:ref/InverseSeries]) function discussed in "Composition and Inversion of Power Series".