The Main Idea 

Some differential equations can be solved by assuming the solution has the form of a power series: [latex]y(x) = \sum_{n=0}^{\infty} a_nx^n[/latex]. We find the coefficients [latex]a_n[/latex] by substituting the series into the differential equation and matching coefficients of like powers of [latex]x[/latex].

The Method

Start by assuming [latex]y(x) = \sum_{n=0}^{\infty} a_nx^n[/latex]. Differentiate term by term to get [latex]y'(x) = \sum_{n=1}^{\infty} na_nx^{n-1}[/latex] and [latex]y''(x) = \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2}[/latex]. Substitute these expressions into the differential equation.

Re-indexing and Combining

After substitution, you’ll have multiple summations with different starting indices and powers of [latex]x[/latex]. Re-index the sums so they all have the same power of [latex]x[/latex] and can be combined into a single summation. This typically involves replacing [latex]n[/latex] with [latex]n+k[/latex] for some integer [latex]k[/latex] and adjusting the summation limits accordingly.

Finding the Coefficients

Once combined into a single sum, the equation can only be satisfied if the coefficient of each power of [latex]x[/latex] equals zero. This gives you a recurrence relation—an equation relating [latex]a_n[/latex] to earlier coefficients. Solve this relation to express all coefficients in terms of [latex]a_0[/latex], [latex]a_1[/latex], or other initial values.

Special Applications

Power series solutions appear in many physical applications. The Bessel equation [latex]x^2y'' + xy' + (x^2 - n^2)y = 0[/latex] arises in problems with cylindrical symmetry, such as vibrating drum heads and heat flow in cylinders. These equations often produce solutions involving even or odd powers only, simplifying the pattern of coefficients.