Mathematics > Differential Equations > Series Solutions

Topic Description:

Series solutions are a pivotal method for solving differential equations, particularly when solutions cannot be expressed in simple closed forms. This approach leverages power series, where the solution is expressed as an infinite sum of terms. The method is especially useful in addressing linear differential equations with variable coefficients, where traditional methods (such as separation of variables or integrating factors) fall short.

Power Series Method

To solve a differential equation using series solutions, one typically assumes that the solution \( y(x) \) can be written as a power series about a point \( x = x_0 \):

\[ y(x) = \sum_{n=0}^{\infty} a_n (x - x_0)^n \]

Here, \( a_n \) represents the coefficients of the series that need to be determined. The convergence of this series is crucial and often depends on the nature of the differential equation and the point \( x_0 \) chosen for expansion (common choices are regular points where the coefficients of the differential equation are analytic).

Procedure for Series Solutions

1. Substitution: Substitute the power series representation of \( y(x) \) into the differential equation.
2. Differentiation: Compute the necessary derivatives of the power series term-by-term.
3. Alignment of Terms: Align terms in the resulting expression to collect coefficients of powers of \( (x - x_0)^n \).
4. Recurrence Relation: Derive a recurrence relation for the coefficients \( a_n \) by equating the sum of coefficients of like powers to zero. This step leverages the linearity of the differential equation.
5. Determining Coefficients: Use the recurrence relation to determine the coefficients \( a_n \), often requiring the initial conditions or boundary conditions of the problem for specific solutions.

Example

Consider the second-order differential equation given by:

\[ x^2 y’’ + x y’ + (x^2 - \nu^2) y = 0 \]

Step 1: Assumption
Assume a power series solution centered at \( x = 0 \):

\[ y(x) = \sum_{n=0}^{\infty} a_n x^n \]

Step 2: Derivatives
Calculate the first and second derivatives:

\[ y’(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} \]

\[ y’’(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} \]

Step 3: Substitution
Substitute these into the differential equation:

\[ x^2 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + x \sum_{n=1}^{\infty} n a_n x^{n-1} + (x^2 - \nu^2) \sum_{n=0}^{\infty} a_n x^n = 0 \]

Step 4: Simplifying
Factor each series to align the powers of \( x \):

\[ \sum_{n=2}^{\infty} n(n-1) a_n x^n + \sum_{n=1}^{\infty} n a_n x^n + \sum_{n=0}^{\infty} a_n x^{n+2} - \nu^2 \sum_{n=0}^{\infty} a_n x^n = 0 \]

Combine and re-index terms for a common power of \( x \):

\[ \sum_{n=0}^{\infty} \left[ n(n-1) a_n + n a_n +(a_{n-2} - \nu^2 a_n ) \right] x^n = 0 \]

Step 5: Recurrence Relation
Set the coefficient of each power of \( x \) to zero to find a recurrence relation:

\[ a_{n+2} = \frac{(\nu^2 - n^2) a_n}{(n+2)(n+1)} \]

This recurrence relation allows for the determination of all coefficients \( a_n \), constructing the series solution.

Application

Series solutions are extensively used in physical sciences and engineering. One notable example is Bessel’s equation, crucial in describing wave propagation, heat conduction, and static potentials. The method’s versatility makes it indispensable in the toolset of applied mathematics, ensuring solutions in contexts where numerical methods are computationally intensive or analytically intractable.

In summary, series solutions provide a systematic approach to solving differential equations by leveraging the power series expansions, enabling solutions in cases where traditional methods fail. Through careful manipulation and understanding of series, highly accurate and insightful solutions can be obtained.