Calculator Inputs
This solver uses a power series around the ordinary point x = 0 for the differential equation y'' + P(x)y' + Q(x)y = R(x).
Example data table
Use this sample to test the solver. It represents y'' − y = 0 with y(0) = 1 and y'(0) = 0, which produces the cosh(x) series.
| Field | Example value | Meaning |
|---|---|---|
| P(x) coefficients | 0 | No y' term contribution. |
| Q(x) coefficients | -1 | Represents −y in the equation. |
| R(x) coefficients | 0 | Homogeneous right side. |
| a₀, a₁ | 1, 0 | Initial series coefficients. |
| Terms | 12 | Computes coefficients through a₁₁. |
| Expected pattern | 1 + x²/2! + x⁴/4! + ... | Even-powered solution matching cosh(x). |
Formula used
Write the differential equation as:
Expand each part around x = 0:
Matching coefficients of xⁿ gives the recurrence:
The calculator starts from a₀ = y(0) and a₁ = y'(0), then computes higher coefficients one by one. It also evaluates the truncated series, its derivatives, and a residual check at the chosen x value.
How to use this calculator
- Enter coefficients for P(x), Q(x), and R(x) as comma or space separated lists.
- Set a₀ and a₁ from the initial conditions of your differential equation.
- Choose how many series terms you want to compute.
- Enter an evaluation point x and the graph interval you want to inspect.
- Click Compute Power Series to generate coefficients, approximations, and the Plotly graph.
- Use the CSV and PDF buttons to export the report after results appear.
FAQs
1) What type of equation does this calculator solve?
It solves second order linear differential equations written as y'' + P(x)y' + Q(x)y = R(x), using a power series around x = 0.
2) What do the coefficient lists represent?
They represent the series coefficients of P(x), Q(x), and R(x). For example, entering 3, -2, 1 means 3 − 2x + x².
3) Why are a₀ and a₁ required?
They are the starting coefficients of the solution series. In ordinary-point problems, they correspond to y(0) and y'(0).
4) Is this an exact symbolic solver?
No. It computes a truncated power series numerically. More terms usually improve local accuracy near the expansion point.
5) What does the residual check mean?
It measures how closely the truncated series satisfies the differential equation at the selected x value. Smaller residuals are better.
6) Why can accuracy drop away from x = 0?
Power series are local approximations. When x moves farther from the expansion point, truncation error can grow and convergence may weaken.
7) Can I solve homogeneous and nonhomogeneous equations?
Yes. Set R(x) = 0 for homogeneous equations, or enter the right-side series coefficients for nonhomogeneous cases.
8) What is the best number of terms to use?
Start with 10 to 16 terms, then increase gradually. Compare residuals and result stability to judge whether more terms help.