AcademyDifferential Equations
Academy
Series Solutions
Level 1 - Calculus topic page in Differential Equations.
Principle
Series solutions represent an unknown function as a power series and determine coefficients by substituting into a differential equation.
Notation
\(y=\sum a_nx^n\)
power series form of the solution
\(a_n\)
coefficient to determine
\(y',y''\)
derivatives of the series
The Core Method
Assume a power series, differentiate term-by-term, substitute into the equation, align powers of \(x\), and equate coefficients.
Power series trial
\[y=\sum_{n=0}^{\infty}a_nx^n\]
The coefficient equations usually produce a recurrence relation for \(a_n\).
Worked Cases
Question
If \(y=\sum_{n=0}^{\infty}a_nx^n\), write \(y'\).
Answer
Differentiate term-by-term: the derivative of \(a_nx^n\) is \(na_nx^{n-1}\). The \(n=0\) term differentiates to zero, so \(y'=\sum_{n=1}^{\infty}na_nx^{n-1}\).
Examples
Question
Why are indices shifted in series solutions?
Answer
After differentiating, powers such as \(x^{n-1}\) appear. To compare coefficients with powers \(x^n\), rewrite the sum using a new index. This makes every term use the same power of \(x\), so matching coefficients is possible.