Calculator Inputs
Use explicit multiplication such as 2*x or x*y. Supported functions include sin, cos, exp, log, sqrt, and powers like x^2.
Compute RK2, RK3, and RK4 solutions precisely. Track k-values, exact comparisons, and interval behavior effortlessly. Export clean tables and visualize numerical solution paths instantly.
Use explicit multiplication such as 2*x or x*y. Supported functions include sin, cos, exp, log, sqrt, and powers like x^2.
This sample uses y' = x + y, x₀ = 0, y₀ = 1, h = 0.2, xₙ = 1, and exact solution 2*exp(x)-x-1.
| Method | Equation | x₀ | y₀ | h | xₙ | Approx Final y | Exact Final y |
|---|---|---|---|---|---|---|---|
| RK4 | y' = x + y | 0 | 1 | 0.2 | 1 | 3.436502 | 3.436564 |
Solve the initial value problem y' = f(x, y) with starting condition y(x₀) = y₀.
k1 = f(x_n, y_n)
k2 = f(x_n + h, y_n + h*k1)
y_(n+1) = y_n + (h/2) * (k1 + k2)
k1 = f(x_n, y_n)
k2 = f(x_n + h/2, y_n + (h/2)*k1)
k3 = f(x_n + h, y_n - h*k1 + 2*h*k2)
y_(n+1) = y_n + (h/6) * (k1 + 4*k2 + k3)
k1 = f(x_n, y_n)
k2 = f(x_n + h/2, y_n + (h/2)*k1)
k3 = f(x_n + h/2, y_n + (h/2)*k2)
k4 = f(x_n + h, y_n + h*k3)
y_(n+1) = y_n + (h/6) * (k1 + 2*k2 + 2*k3 + k4)
f(x, y).x₀ and initial value y₀.h and the target point xₙ.It solves first-order ordinary differential equations written as y' = f(x, y). You provide an initial point and a target x-value, and it computes the numerical solution step by step.
RK2 is faster and simpler, RK3 balances speed and accuracy, and RK4 is usually the preferred general-purpose choice because it offers strong accuracy for many practical problems.
Yes. Enter the exact solution in terms of x only. The calculator will display exact values and absolute errors at each computed step.
Use explicit multiplication like 2*x and x*y. Supported forms include sin(x), cos(x), exp(x), log(x), sqrt(x), and powers like x^2.
If the interval does not divide evenly by the chosen step size, the calculator automatically shortens the last step so the solution ends exactly at the requested target x-value.
Usually yes, but smaller steps also increase computation time and table length. Very small steps may be unnecessary when RK4 already gives enough precision for your problem.
The chart plots the numerical solution against x. If you enter an exact solution, the graph also shows the exact curve so you can visually compare both paths.
Yes. If the target x-value is less than the starting x-value, the calculator automatically reverses the step direction and computes the solution backward.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.