Runge-Kutta 4 single-step calculator

Enter the derivative as f(x, y), the starting coordinates, and a nonzero step size. The result shows k1, k2, k3, and k4 as slope estimates, the intermediate points where they are evaluated, their weighted average, and the final RK4 approximation. That complete trail makes the calculator useful for checking homework, validating a numerical solver, or documenting one step without hiding the arithmetic.

Describe the initial-value problem clearly

Start with a first-order equation written as y' = f(x, y). In the expression field, enter only the right-hand side, using x and y as the independent and dependent variables. For example, if y' = x + y, enter x + y. The parser supports ordinary arithmetic, parentheses, powers written with ^, the constants pi and e, and common functions such as sin, cos, exp, log, and sqrt. Then provide the known point (x, y) and the step size h. A positive h advances to the right, while a negative h performs the same RK4 construction backward. The step size cannot be zero because that would not advance the solution. All numeric inputs and every function value must remain finite. Multiplication must be explicit, so write 2*x rather than 2x. This precise input format prevents an equation label or implicit multiplication from being interpreted in an unintended way and keeps the four evaluations reproducible.

Follow the four slope estimates

The calculator reports k1 through k4 as slopes, following the convention k1 = f(x, y), rather than as increments already multiplied by h. First, k1 uses the initial point. The second estimate uses the midpoint x + h/2 and predicts its y-coordinate with h*k1/2. The third estimate stays at the midpoint but replaces that prediction with h*k2/2, which corrects the direction using the second slope. Finally, k4 evaluates the derivative at x + h with a full-step y prediction based on k3. The returned k2_point, k3_point, and k4_point fields expose those evaluation coordinates directly. This matters because textbooks sometimes use the same k symbols for h times each slope. If your source follows that alternative convention, multiply each reported k value by h before comparing individual stages. The final y approximation is unchanged when the corresponding formula is used consistently.

Interpret the weighted update and its limits

RK4 combines the slopes as (k1 + 2*k2 + 2*k3 + k4)/6, giving the two midpoint estimates twice the weight of the endpoint estimates. The calculator returns this quantity as weighted_slope, then computes y_next = y + h*weighted_slope and x_next = x + h. One RK4 step is fourth-order accurate in the standard local sense for a sufficiently smooth derivative, but that statement is not a guarantee that any chosen step size is small enough. A rapidly changing, singular, discontinuous, or stiff problem may require a much smaller h or a method with error control. This tool deliberately performs exactly one fixed step; it does not estimate truncation error, adapt the step size, or solve across an interval. To assess sensitivity, compare one step of size h with two successive steps of size h/2 in a full solver. Use the displayed stages to locate differences caused by expression entry, rounding, or slope convention.