Series Solutions of Differential Equations

Learning Objectives

  • Use power series to solve first-order and second-order differential equations.

problem-solving strategy: finding power series solutions to differential equations

  1. Assume the differential equation has a solution of the form [latex]y(x)=\displaystyle\sum_{n=0}^\infty a_nx^n[/latex].
  2. Differentiate the power series term by term to get [latex]y^\prime(x)=\displaystyle\sum_{n=0}^\infty na_nx^{n-1}[/latex] and [latex]y^{\prime\prime}(x)=\displaystyle\sum_{n=2}^\infty n(n-1)a_nx^{n-2}[/latex].
  3. Substitute the power series expressions into the differential equation.
  4. Re-index sums as necessary to combine terms and simplify the expression.
  5. Equate coefficients of like powers of [latex]x[/latex] to determine values for the coefficients [latex]a_n[/latex] in the power series.
  6. Substitute the coefficients back into the power series and write the solution.

Example: series solutions to differential equations

Find a power series solution for the following differential equations.

  1. [latex]y^{\prime\prime}-y=0[/latex]
  2. [latex](x^2-1)y^{\prime\prime}+6xy^\prime+4y=-4[/latex]

try it

Find a power series solution for the following differential equations.

  1. [latex]y^{\prime}+2xy=0[/latex]
  2. [latex](x+1)y^\prime=3y[/latex]

Watch the following video to see the worked solution to the above Try It

We close this section with a brief introduction to Bessel functions. Complete treatment of Bessel functions is well beyond the scope of this course, but we get a little taste of the topic here so we can see how series solutions to differential equations are used in real-world applications. The Bessel equation of order [latex]n[/latex] is given by

[latex]x^2y^{\prime\prime}+xy^\prime+(x^2-n^2)y=0[/latex].

This equation arises in many physical applications, particularly those involving cylindrical coordinates, such as the vibration of a circular drum head and transient heating or cooling of a cylinder. In the next example, we find a power series solution to the Bessel equation of order 0.

Example: power series solution to the bessel equation

Find a power series solution to the Bessel equation of order [latex]0[/latex] and graph the solution.

try it

Verify that the expression found in Example “Power Series Solution to the Bessel Equation” is a solution to the Bessel equation of order 0.