Differential Equations

Author

Apurva Nakade

Published

July 13, 2026

Numerical methods for ordinary differential equations approximate the solution of an initial value problem \(y' = f(t, y)\), \(y(t_0) = y_0\) by stepping forward in time, using only information already known at or before each step.

Adaptive Methods for ODEs

Computational math

Differential equations

Runge-Kutta methods

Adaptive step size

Initial value problems

The classical RK4 method in Explicit Methods advances with a fixed step size \(h = b/n\) everywhere — spending the same effort even where the solution is nearly linear. RKF45…

Butcher Tableau Explorer

Computational math

Differential equations

Runge-Kutta methods

Butcher tableau

An \(s\)-stage single-step Runge-Kutta method is specified by its Butcher tableau: nodes \(c_i\), coefficients \(a_{ij}\) (\(1 \le i, j \le s\)), and weights \(b_i\)…

Explicit Methods for ODEs

Computational math

Differential equations

Runge-Kutta methods

Initial value problems

Given an initial value problem \(y' = f(t, y)\), \(y(0) = y_0\), an explicit method advances the solution step by step: each new value \(y_{i+1} \approx y(t_{i+1})\) is…

Implicit Methods for ODEs

Computational math

Differential equations

Implicit methods

Initial value problems

Given an initial value problem \(y' = f(t, y)\), \(y(0) = y_0\), Euler’s method (forward/explicit Euler) uses the slope at the start of the step: