A differential equation calculator that numerically solves first-order ordinary differential equations of the form dy/dx = f(x, y) using Euler or Runge–Kutta 4 (RK4) methods, and renders interactive plots using Plotly.js for immediate visualization and export.

How to use the Differential Equation Calculator

Quick overview

This Differential Equation Calculator is a compact, WordPress-ready custom code widget that solves first-order ODEs of the form dy/dx = f(x, y) numerically and displays the solution in an interactive Plotly.js chart. It accepts a function expression, initial conditions, an x-range and step count, and offers both Euler and RK4 methods for computation. The interface is intentionally simple for seamless embedding between standard WordPress sidebars and fits the usual content width.

Why this tool is useful

Users often need a fast numerical solution and visual intuition for how solutions behave — whether for homework, prototyping, or demonstrations. Analytical closed-form solutions are not always available, so a robust numerical solver paired with an interactive Plotly.js graph helps you:

• Inspect trajectories quickly,
• Compare solver accuracy (Euler vs RK4),
• Export solution points for further analysis,
• Embed an attractive, responsive visualization into a WordPress post or page.

Interface and inputs (what each control does)

Expression input

Enter the right-hand side of the ODE in JavaScript/math-style syntax, e.g. -2*x*y + x or sin(x) - y^2. Use variables xand y. The calculator uses math.js to parse expressions, so common math functions (sin, cos, exp, log, etc.) work as expected.

Initial conditions and domain

• x0 — initial x value where y is known.
• y0 — initial y value at x0.
• End x — the final x value to calculate up to. The solver integrates from x0 to End x.
• Steps (n) — number of steps for the numerical method. Larger n gives finer resolution and typically better accuracy, at the cost of more computation.

Method selector

Choose between:

• Euler — simplest explicit method (fast but low accuracy for stiff or rapidly changing problems).
• RK4 (Runge–Kutta 4) — a widely used, robust single-step method that balances accuracy and performance for general-purpose use.

Optional parameters

You can supply simple parameter replacements like a=2; b=3 in a notes field if you prefer to reuse expressions with constants (e.g., a*x - b*y).

Buttons

• Solve & Plot — computes numeric solution and renders it with Plotly.js.
• Reset — clears fields and plot.
• Export CSV — downloads the computed (x, y) pairs as a CSV file.

How the solver works (brief technical explanation)

The tool compiles your expression using math.js into a function f(x,y). For RK4, the solver computes four intermediate slopes (k1, k2, k3, k4) per step and advances the solution using a weighted average of these slopes. RK4 is fourth-order accurate in step size and usually gives very good results for smooth problems. Euler’s method takes a single slope evaluation per step and advances linearly; it’s simpler but less accurate and more sensitive to step size.

Plotly.js is used to draw an interactive chart with zoom, pan, hover tooltips, and the ability to save the plot as an image. The plot background is white to match your site styling and readability preferences.

Embedding into WordPress

• Insert the HTML code into a Custom HTML block (for Gutenberg) or paste into an HTML widget inside your theme’s custom code area.
• The file is sized with max-width: 760px so it will sit comfortably between sidebars on most themes. It is responsive and scales down for mobile.
• Ensure external scripts (Plotly.js and math.js) are reachable — the code uses their CDN versions. If your site restricts external scripts, you can host the libraries locally and update the script tags accordingly.

Practical tips & common workflows

Choosing step count and method

• Start with RK4 and n = 200 for typical ranges. Increase n for finer detail or if the solution appears jagged.
• If the function has rapid oscillations or stiff behavior, reduce step size (increase n) and consider more advanced solvers (this widget focuses on RK4/Euler).

Validation

• Compare RK4 vs Euler for the same problem: if they match closely, your solution is probably well-resolved.
• For ODEs with known analytical solutions, plot both the numerical solution (from the widget) and the analytical formula externally to validate.

Saving and exporting

• Use the Export CSV button to download data for plotting elsewhere or for use in spreadsheets and numerical analysis tools.
• Use Plotly’s built-in menu to save a PNG of the chart directly.

Accessibility & styling

• The calculator uses readable font sizes, simple contrast, and a white background for clarity.
• Controls are keyboard-accessible. You can style further to match your theme (CSS selectors are straightforward to override).

Example usage

1. Enter dy/dx = -2*x*y + x in the expression box.
2. Set x0 = 0, y0 = 1, End x = 5, Steps = 200, choose RK4.
3. Click Solve & Plot. The plot will render the numeric solution of y(x).
4. If you want more detail, raise Steps to 1000 and compare the result.

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FAQ

Q: Which differential equations can this calculator solve?
A: It solves first-order ordinary differential equations of the form dy/dx = f(x,y). For higher-order ODEs, you can reduce them to a system of first-order ODEs manually, but the current interface supports only a single scalar equation.

Q: Is the expression syntax JavaScript?
A: It uses math.js syntax (very similar to JavaScript math expressions). Use sin(x), cos(x), exp(x), log(x), x^2 or x**2where appropriate. Use variables x and y.

Q: How do I improve accuracy?
A: Use the RK4 method and increase the number of steps. If you need very high accuracy or stiffness handling, consider a dedicated ODE solver backend (e.g., Sundials or SciPy) or extend the widget with adaptive-step methods.

Q: Can I style the widget to match my site?
A: Yes — the CSS is minimal and uses a white background. You can override styles in your theme or custom CSS. The container element has the class .deq-container.

Q: Can the widget handle discontinuities or singularities?
A: Numerical methods struggle with singularities and discontinuities. If f(x,y) is undefined at points in the interval, the solver will likely fail or produce NaNs. Break the interval into safe subintervals or handle piecewise definitions manually.

Q: Are there any security concerns with parsing expressions?
A: The widget uses math.js to parse mathematical expressions, not raw eval on arbitrary JS. Still, sanitize inputs and consider hosting math.js locally if your environment restricts external CDNs.