Implicit Euler ODE Solver
Use backward, or implicit, Euler for a first-order initial-value problem. Unlike forward Euler, the new value appears inside f at xₙ₊₁, so each step requires a nonlinear solve.
Expression syntax: use explicit multiplication such as 2*y. Supported functions include sin, cos, tan, exp, ln, sqrt and abs.
How this tool works
Implicit Euler solves yₙ₊₁ = yₙ + h f(xₙ₊₁,yₙ₊₁). The calculator uses Newton iteration with a numerically estimated derivative for the scalar nonlinear equation.
Example
For y′ = −25y, backward Euler remains stable for step sizes that can make forward Euler oscillate or diverge, although it is only first-order accurate.
Common questions
Why use a first-order implicit method?
Backward Euler is A-stable and strongly damps rapidly decaying modes, making it a useful baseline for stiff equations.
Can Newton iteration fail?
Yes. Strong nonlinearities or a large step can make the implicit equation difficult to solve. Reducing h usually improves convergence.