What is the theoretical justification for assuming that a po

What is the theoretical justification for assuming that a power series will provide a solution to a differential equation? (In other words, explain why power series methods for solving linear differential equations usually work, at least for most problems of practical interest.)

Solution

There are differential equations of great importance in higher mathematics and engineering that cannot be solved in terms of elementary functions.

In such cases one must turn to approximate methods such as power series. The solutions to many differential equations are expressible in terms of a power series of the form

1)        y = a₀ + a₁x + a₂x² + a₃x³ + ....... .