What makes an ODE separable?

An ODE is separable if the right-hand side can be written as a product of a function of x only and a function of y only: dy/dx = f(x)·g(y). This allows the equation to be split and integrated separately on each side.

How accurate is the RK4 method?

RK4 is a 4th-order method, meaning the local truncation error is O(h^5) and the global error is O(h^4). For most smooth ODEs, a step size of 0.1 or smaller gives very accurate results.

What syntax can I use in the expressions?

Expressions are parsed by math.js. You can use standard operators (+, -, *, /, ^), functions (sin, cos, exp, log, sqrt, abs), and constants (pi, e). Use x in f(x) and y in g(y).

Why does the solver stop early or show a blowup warning?

Some ODEs have solutions that grow to infinity in finite time (a "blowup" or singularity). The solver stops and warns you when |y| exceeds 1,000,000 to avoid meaningless results.

Can I solve non-separable ODEs with this tool?

No. This tool is specifically for separable ODEs of the form dy/dx = f(x)·g(y). For non-separable ODEs you would need a general-purpose ODE solver.

Are separable ODEs always first-order?

Yes — the separable form dy/dx = f(x)·g(y) is first-order by definition (it contains only the first derivative dy/dx). Higher-order ODEs are not separable in this sense; some can still be reduced to a first-order system that you can solve with our generic Runge-Kutta calculator.

Separable ODE Solver

Name: Separable ODE Solver
Availability: InStock
Author: Nham Vu

Numerically solve separable first-order ODEs dy/dx = f(x)·g(y) using the RK4 method and plot the solution curve.

Presets:

ODE: dy/dx = f(x) · g(y)

f(x) expression

g(y) expression

x₀ (initial x)

y₀ (initial y)

x min

x max

Step h

Solution Table

x	y
Press "Solve ODE" to compute the solution.

Solution Curve

Summary

Numerically solve separable first-order ODEs dy/dx = f(x)·g(y) using the RK4 method and plot the solution curve.

How it works

Write the ODE in separable form dy/dx = f(x)·g(y) and type the two factors into the f(x) and g(y) inputs (math.js parses sin, cos, exp, log, sqrt, ^, etc.).
Set the initial point (x0, y0), the x range [xmin, xmax], and a step size h — smaller h gives more accuracy.
For each step the tool computes F(x, y) = f(x)·g(y) and four RK4 slopes k1 = h·F(xn, yn), k2 = h·F(xn + h/2, yn + k1/2), k3 = h·F(xn + h/2, yn + k2/2), k4 = h·F(xn + h, yn + k3).
It advances with yn+1 = yn + (k1 + 2k2 + 2k3 + k4) / 6 — 4th-order accuracy with global error O(h^4).
Solution values are listed step-by-step and plotted as y(x); a blowup ( |y| > 1e6 ) stops the run and warns you of a singularity.

Use cases

Solve population growth models where dy/dx = r·y.
Analyze exponential decay problems in physics or chemistry.
Explore solution curves for dy/dx = x/y (circular-arc families).
Verify hand-calculated separable ODE solutions numerically.
Visualize how changing initial conditions shifts the solution curve.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu