Classical fourth-order Runge-Kutta for dy/dx = f(x, y): step table, k1–k4 stage details, final estimate, and optional exact-solution error column.
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Key facts
Category
Math & Numbers
Input types
text, number
Output type
text
Sample coverage
4
API ready
Yes
Overview
The Runge-Kutta RK4 Solver calculates high-accuracy numerical solutions for first-order initial value problems defined as dy/dx = f(x, y). Using the classical fourth-order Runge-Kutta algorithm, it computes intermediate slopes (k1 through k4) at each step to generate an iteration table, final estimate, and optional error comparison against an exact analytical solution.
When to use
When you need to solve a first-order ordinary differential equation numerically without writing custom code.
When verifying analytical calculus solutions or homework problems against RK4 iteration tables.
When evaluating the numerical convergence and error bounds of an initial value problem across varying step counts.
How it works
1Enter the differential function f(x, y), initial coordinates (x₀, y₀), and target x value.
2Specify the number of steps to define the step size h = (xEnd - x0) / steps, and optionally set precision decimals or an exact y(x) formula.
3The solver computes four weighted slopes (k1, k2, k3, k4) for every interval using the standard RK4 update formula y_{n+1} = y_n + (h/6) * (k1 + 2k2 + 2k3 + k4).
4Review the resulting iteration table showing x_i, y_i, stage calculations for initial steps, and absolute error values.
Use cases
Simulating standard decay and growth differential equations where closed-form solutions require numerical verification.
Generating step-by-step intermediate stage values (k1–k4) for differential equation coursework and demonstrations.
Approximating non-linear differential equations that lack straightforward closed-form elementary antiderivatives.
Examples
1. Exponential Growth Integration
Applied Mathematics Student
Background
Studying numerical integration methods for autonomous ODEs and verifying convergence on dy/dx = y.
Problem
Calculate y(1) given y(0) = 1 in 10 steps and compare the numerical result directly with the exact solution e^x.
How to use
Set f(x, y) to 'y', x₀ to 0, y₀ to 1, target x to 1, steps to 10, and exact solution to 'e^x'.