First-order linear ODE

The method

A worked example

For y′ = y and y(0) = 1, y(x) = exp(x), so y(1) ≈ 2.7182818285. For a = 0, y = y₀ + b(x − x₀).

Assumptions & limits

Constant real coefficients only. Stable expm1 evaluation handles small nonzero a; only a = 0 uses the linear solution. Extreme exponentials, overflow, and cancellation between physical terms can still require rescaling. This is not a general symbolic ODE solver. Full double precision between operations; displayed values use up to 10 decimal places. Numerical outputs above 10¹⁰⁰ and nonfinite function evaluations are rejected.

Calculation source: Math Workbench: constant-coefficient ODE solver with stable evaluation for small coefficients.

Explore Math Workbench ↗