Quick Calculators

Series Solution of ODEs Finder: Advanced Differential Equations

Series Solution of ODEs Solver

Series Solution Solver

Find power series solutions $y(x) = \sum a_n x^n$ for second-order linear ODEs around $x=0$.

ODE Definition

$$ y” + P(x)y’ + Q(x)y = 0 $$

Coeff of y’. Use ‘x’. Example: -2x

Coeff of y. Use ‘x’. Example: 1

Presets: Airy Hermite Harmonic

Initial Conditions at x=0

Power Series Expansion

Approximation
$$ y(x) \approx 1 + x + \frac{1}{6}x^3 + … $$
Coefficients ($a_n$):

Convergence Plot

Plot Range: x ∈ [-4, 4]
Red: Series Approximation | Blue Dashed: Numerical Solution (RK4)

Power Series Method Explained

When a differential equation cannot be solved using elementary functions, we assume the solution can be represented as a Power Series: $$ y(x) = \sum_{n=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots $$

The Method:
1. Substitute the series and its derivatives ($y’, y”$) into the ODE.
2. Shift indices so all powers of $x$ match (usually $x^n$).
3. Set the total coefficient of each power of $x$ to zero to find a Recurrence Relation.

This tool automates finding the coefficients $a_n$. $a_0$ and $a_1$ are determined by the initial conditions $y(0)$ and $y'(0)$. Subsequent terms ($a_2, a_3, \dots$) are calculated recursively.

Ordinary Points

This method is valid around an Ordinary Point $x=0$, meaning $P(x)$ and $Q(x)$ are analytic at $x=0$. If $P(x)$ or $Q(x)$ have singularities (like $1/x$), we would need the method of Frobenius, which allows for fractional or negative powers.

FAQ

What is the Radius of Convergence?
The series solution converges only within a certain interval $(-R, R)$. The radius $R$ is typically the distance from the expansion point ($x=0$) to the nearest singularity of the ODE in the complex plane. Outside this range, the polynomial approximation (Red line) will diverge wildly from the true solution (Blue dashed line).
Why does the graph diverge at the edges?
Polynomials (truncated series) always go to $\pm \infty$ as $x \to \pm \infty$. A finite number of terms can only approximate the function locally near $x=0$. Increasing the “Series Order” improves accuracy further out, but eventually, it will always fail for non-polynomial solutions.

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