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

You can view the transcript for “CP 7.22a” here (opens in new window).
You can view the transcript for “CP 7.22b” here (opens in new window).

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.