Functions as Polynomials
If we can't solve a DE with known functions (like \(\sin\) or \(e^x\)), we can assume the solution is an infinite polynomial: a Power Series.
\[y(x) = \sum_{n=0}^{\infty} a_n x^n\]
We substitute this into the DE and find a "Recurrence Relation" that allows us to calculate each coefficient \(a_n\) from the previous ones.
Worked Examples
Example 1: Solving \(y' = y\)
While we know the answer is \(e^x\), let's use a power series.
- \(y = a_0 + a_1 x + a_2 x^2 + \dots\)
- \(y' = a_1 + 2a_2 x + 3a_3 x^2 + \dots\)
- Set them equal: \(a_1 = a_0\), \(2a_2 = a_1 \implies a_2 = a_0/2\), \(3a_3 = a_2 \implies a_3 = a_0/6\).
- The pattern is \(a_n = a_0 / n!\).
- Result: \(y = a_0 \sum \frac{x^n}{n!} = a_0 e^x\).
The Bridge to Quantum Mechanics
Power series are the ultimate weapon for solving the Schrödinger Equation in complex potentials. The Harmonic Oscillator and the Hydrogen Atom are both solved using this method. The "polynomial" part of the wavefunction (like the Hermite Polynomials) is what determines the shape and nodes of the electron orbitals. Without power series, we could never prove that electrons are restricted to specific energy levels.