# Runge-Kutta RK4 Solver (dy/dx = f(x, y)) 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. > Canonical page: https://elysiatools.com/en/tools/runge-kutta-rk4-solver - **Category:** Math & Numbers - **Keywords:** runge kutta, rk4, fourth order runge kutta, numerical ode, numerical integration, initial value problem, dy/dx, step size, ode45, calculus ## 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. ## Inputs - **f(x, y) in dy/dx = f(x, y)** (text): Right-hand side in terms of x and y (functions: sin, cos, exp, ln, sqrt, …). Write products with *, e.g. x*y. - **x₀ (initial x)** (number): e.g. 0 - **y₀ = y(x₀)** (number): e.g. 1 - **Target x** (number): Must be greater than x₀. - **Number of Steps n** (number): h = (target x − x₀)/n. More steps → smaller error, O(h⁴) globally. - **Exact Solution y(x) (optional)** (text): Exact solution in x, used to add an error column. Leave empty to skip. - **Decimal Places** (number) ## 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 - Enter the differential function f(x, y), initial coordinates (x₀, y₀), and target x value. - Specify the number of steps to define the step size h = (xEnd - x0) / steps, and optionally set precision decimals or an exact y(x) formula. - The 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). - Review 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. ## Frequently asked questions ### What is the global truncation error of the RK4 method? The RK4 method has a global truncation error of O(h⁴), meaning halving the step size h reduces numerical error by approximately a factor of 16. ### How should mathematical expressions for f(x, y) be formatted? Use explicit arithmetic operators like * for multiplication (e.g., -2*x*y) and standard functions such as sin, cos, exp, ln, and sqrt. ### What does providing an exact solution formula do? Entering an optional exact expression y(x) evaluates the analytical value at each step and outputs an absolute error column (|y_exact - y_approx|). ### Can the target x value be smaller than x₀? No, the target x value must be greater than the initial x₀. ### What is the maximum number of steps allowed? 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