Euler Method for a 2D System Calculator

Enter both derivative expressions, the initial state, a fixed step size, and the number of steps. The result shows each simultaneous update, including the slopes evaluated at the preceding point, so you can inspect the numerical path instead of receiving only a final answer. It is useful for coursework, quick model checks, and reproducible calculations where a transparent first-order method is appropriate.

Describe the coupled initial-value problem

Write the first derivative as an expression for dy/dx and the second as an expression for dz/dx. Both expressions may use the independent variable x and the two dependent variables y and z. For example, a system can use y + z for the first derivative and x - y for the second. Multiplication must be explicit, so enter 2*y rather than 2y. The parser supports parentheses, decimal and scientific-notation numbers, the operators +, -, *, /, and ^, the constants pi and e, and the functions sin, cos, tan, exp, log, sqrt, and abs. Then provide x0, y0, and z0 to identify one initial state unambiguously. The two equations are coupled because either derivative can depend on either dependent variable. The calculator evaluates both slopes from the same current state before updating either variable, which is essential: using an already updated y while calculating z would no longer be the standard explicit Euler method and could produce a different trajectory.

Understand each fixed Euler update

For a step size h and current point (x_n, y_n, z_n), the calculator evaluates f_n = f(x_n,y_n,z_n) and g_n = g(x_n,y_n,z_n). It then applies y_(n+1) = y_n + h*f_n, z_(n+1) = z_n + h*g_n, and x_(n+1) = x_n + h. The output begins with step zero, which records the supplied initial state, and then lists every computed point. Each later row includes dy_dx and dz_dx, the two slopes used to move from the preceding point into that row. This layout makes hand-checking straightforward: multiply each listed slope by the fixed step size and add it to the prior dependent value. A negative step size is allowed when you need to approximate the solution toward smaller x values. The number of iterations is deliberately separate from h, so the endpoint is x0 + steps*h and remains fully predictable without hidden endpoint rounding or a shortened final interval.

Choose a step size and interpret the approximation

Euler's method is a first-order numerical approximation, not a symbolic solver and not an error guarantee. A smaller absolute step size often reduces discretization error for a smooth, well-behaved system, but it also requires more iterations to cover the same interval. A practical check is to run the same problem again with half the step size and twice as many steps, then compare states at matching x values. Large disagreement indicates that the coarser calculation is not yet stable enough for your purpose. Some systems are stiff, singular, discontinuous, or highly sensitive, and explicit Euler may behave poorly even when the entered formulas are correct. The calculator stops rather than returning misleading JSON if a derivative or update becomes infinite or not-a-number. Use the displayed path to learn, audit a worked exercise, or make an early estimate. For safety-critical simulation or a result requiring controlled accuracy, use an adaptive higher-order method and validate the mathematical model, tolerances, and units independently.