## Series Solutions of Differential Equations

Many differential equations do not have analytical solutions – solutions expressed in terms of elementary functions. In these circumstances, a numerical solution must be found. One possible numerical solution is in terms of a series expansion.

To find the series expansion we assume a solution of the form and substitute into the differential equation, collecting together and equating powers of If there are any elementary functions in the equation, these are also expanded in powers of The solution is then found iteratively using the boundary or initial condition(s). If there is one condition, or then or respectively is found first, then the other found by iteration. For higher order differential equations, equations in the must be formed and solved then the other found by iteration.

Example: Solve with and Assume a series solution of the form then and The equation to be solved becomes We have to collect like powers of but must first re - index the summations so that and become To do this write  The equation becomes We can take all the summations inside a single summation to give Equate each coefficient of to zero, since the right hand side is zero.  But and so and The solution is with higher order terms defined with a recurrence relation in terms of coefficients of Example with and Assume a series solution of the form then and The equation to be solved becomes We have to collect like powers of but must first re - index the summations so that and become To do this write  The equation becomes We can take all the summations inside a single summation to give Equate each coefficient of to zero, since the right hand side is zero.  But and so and The solution is with higher order terms defined with a recurrence relation in terms of coefficients of  