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 chart below shows each accepted step’s increment \(\Delta y_i = y_{i+1}^{(5)} - y_i\) across its time interval \([t_i, t_{i+1}]\), with a semi-transparent overlay of the…

Butcher Tableau Explorer

Computational Math

Differential Equations

Runge-Kutta Methods

Butcher Tableau

An \(s\)-stage Runge-Kutta method is specified by a Butcher tableau of nodes \(c_i\), coefficients \(a_{ij}\), and weights \(b_i\). Enter coefficients as integers…

Explicit Methods for ODEs

Computational Math

Differential Equations

Runge-Kutta Methods

Initial Value Problems

Given \(y' = f(t, y)\), \(y(0) = y_0\), an explicit method computes each \(y_{i+1} \approx y(t_{i+1})\) directly from already-known values. Compare Euler’s method (one slope…

Implicit Methods for ODEs

Computational Math

Differential Equations

Implicit Methods

Initial Value Problems

Implicit Euler (backward Euler) uses the slope at the end of the step, \(y_{i+1} = y_i + h\, f(t_{i+1}, y_{i+1})\), which requires solving for \(y_{i+1}\) at each step (this…