Runge-Kutta 2 ODE Solver Calculator

Enter the derivative expression, the known point x0 and y0, an endpoint, and a fixed number of steps. You can use the midpoint, Heun, or Ralston second-order formula. The result reports the common step size, both derivative slopes from every step, each weighted increment, and the final approximation, so the numerical process remains transparent and reproducible.

Enter the differential equation and interval

Put only the right-hand side f(x,y) in the equation field. For y' = x + y, enter x + y. The expression supports decimal and scientific-notation numbers, the variables x and y, the constants pi and e, parentheses, addition, subtraction, explicit multiplication, division, powers, and the functions exp, log, sin, cos, and sqrt. Write 2*x rather than 2x, and place function arguments inside parentheses. The initial condition is defined by x0 and y0, meaning that y(x0) equals y0. target_x marks the other end of the interval. It may be above or below x0, so forward and backward integration both work. Finally, choose the exact number of steps. The calculator divides the complete interval into that many equal pieces; it does not interpret the number as a maximum or adjust it while solving. A malformed expression, unsupported name, missing numeric value, or calculation outside the real finite domain produces an input error instead of a partially trustworthy table.

Choose and read an RK2 formula

Every included method is a two-stage, second-order explicit Runge-Kutta formula. First, the solver evaluates k1 = f(x,y) at the current point. It then uses k1 to predict an intermediate location and evaluates k2 there. The midpoint method places that stage halfway through the step and uses k2 as the final slope. Heun's method predicts at the end of the step and averages k1 with k2; it is also called the improved Euler or explicit trapezoid method. Ralston's method places its stage two-thirds through the step and combines one quarter of k1 with three quarters of k2, a choice designed to reduce a common local-error coefficient. The output lists k1, k2, and the resulting increment for each accepted point. Add the increment to the preceding y value to reproduce the displayed next value. Because the stage rules differ, two RK2 variants can return slightly different approximations even though all three have second-order convergence for sufficiently smooth problems.

Assess step size and numerical reliability

The signed step size is (target_x - x0) divided by steps, so increasing the step count reduces the magnitude of every step. For a smooth non-stiff problem, a second-order method generally reduces global error by about a factor of four when the step size is halved, once the calculation is in its asymptotic range. A practical convergence check is therefore to solve with N steps and again with 2N steps, then compare the final y values. Agreement is evidence of stability for that problem, but it is not a rigorous error certificate. Inspect the reported slopes as well: sudden growth, alternating large values, or non-finite arithmetic can indicate that the interval crosses a singularity or that the chosen steps are too coarse. This calculator deliberately uses a fixed count and performs no adaptive error control. It also returns an approximation, not a symbolic solution. Stiff equations, discontinuous derivatives, long integrations, and safety-critical scientific calculations should be checked with an appropriate adaptive solver, domain knowledge, and independent error analysis.