## 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 formand substitute into the differential equation, collecting together and equating powers ofIf there are any elementary functions in the equation, these are also expanded in powers ofThe solution is then found iteratively using the boundary or initial condition(s). If there is one condition,or thenorrespectively is found first, then the otherfound by iteration. For higher order differential equations, equations in themust be formed and solved then the otherfound by iteration.

Example: Solvewithand

Assume a series solution of the formthenandThe equation to be solved becomes

We have to collect like powers ofbut must first re - index the summations so thatand becomeTo do this write

The equation becomes

We can take all the summations inside a single summation to give

Equate each coefficient ofto zero, since the right hand side is zero.

ButandsoandThe solution is

with higher order terms defined with a recurrence relation in terms of coefficients of

Example

withand

Assume a series solution of the formthenandThe equation to be solved becomes

We have to collect like powers ofbut must first re - index the summations so thatand becomeTo do this write

The equation becomes

We can take all the summations inside a single summation to give

Equate each coefficient ofto zero, since the right hand side is zero.

ButandsoandThe solution is

with higher order terms defined with a recurrence relation in terms of coefficients of