Differential Equation Calculator

Solve first-order ordinary differential equations (ODEs) of the form dy/dx = f(x, y) with initial conditions using numerical methods like Euler's or Runge-Kutta (4th order). Visualize the solution curve and compare approximation accuracy.

Initial x (x₀) Starting point of the solution interval

Initial y (y₀) Value of y at x = x₀ (initial condition)

Final x (x₁) Endpoint of the solution interval (x₁ > x₀)

Number of Steps (n) Higher values increase accuracy; typical range: 10–100

Numerical Method

Differential Equation: dy/dx = Enter a valid expression in terms of x and y (use * for multiplication)

Quick Examples:

How to Use This Calculator

Enter the initial condition (x₀, y₀) and target x value (x₁)
Specify the number of steps and choose your numerical method
Input your differential equation using valid math syntax (e.g., x*y, sin(x), exp(y))
Click Calculate to see the approximate y(x₁) and solution graph

Formula Used

y_{n+1} = y_n + h·f(x_n, y_n)   (Euler) 

y_{n+1} = y_n + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)   (Runge-Kutta)

Where:

h = (x₁ - x₀)/n = step size
f(x, y) = right-hand side of dy/dx = f(x, y)
k₁ = f(xₙ, yₙ), k₂ = f(xₙ + h/2, yₙ + h·k₁/2), etc. for Runge-Kutta

Example Calculation

Real-World Scenario:

Modeling radioactive decay where the rate of decay is proportional to the current amount: dy/dx = -k·y, with k = 2.

Given:

Initial x (x₀) = 0
Initial y (y₀) = 1
Final x (x₁) = 2
Steps (n) = 20
Equation: dy/dx = -2*x*y

Calculation:

Using Runge-Kutta 4th order with h = (2–0)/20 = 0.1, we compute 20 iterative steps. At each step, we evaluate four slopes (k₁–k₄) and update y.

Result: y(2) ≈ 0.0183, which closely matches the exact solution y = e^−x² ≈ e⁻⁴ = 0.0183.

Why This Calculation Matters

Practical Applications

Population dynamics in ecology
Heat transfer and thermodynamics
Circuit analysis in electrical engineering

Key Benefits

Approximates solutions when analytical methods fail
Visualizes dynamic system behavior over time
Compares accuracy of different numerical methods

Common Mistakes & Tips

Forgetting explicit multiplication (e.g., writing "xy" instead of "x*y") causes parsing errors. Always use * for multiplication and valid function names like sin, cos, exp, log.

Using fewer than 10 steps may yield poor approximations, especially for stiff equations. Start with 20–50 steps and increase if results seem unstable.

Frequently Asked Questions

No, this calculator is designed for first-order ODEs only. Second-order equations require conversion to a system of first-order equations, which is beyond this tool's scope.

Runge-Kutta (4th order) is significantly more accurate than Euler’s method, especially for nonlinear equations or larger step sizes. Euler’s method is mainly for educational illustration.

The graph plots all intermediate (xₙ, yₙ) points computed during the numerical integration, connecting them with lines to visualize the approximate solution curve from x₀ to x₁.

References & Disclaimer

Educational Disclaimer

This calculator provides numerical approximations for educational purposes only. Results are not guaranteed to be exact or suitable for engineering, scientific, or professional decision-making without expert validation.

References

Runge-Kutta Method - Detailed mathematical explanation from Wolfram MathWorld
Euler Method - Wikipedia’s comprehensive overview of the algorithm and error analysis
Paul's Online Math Notes: ODEs - Introduction to differential equations and solution concepts

Accuracy Notice

Accuracy depends on step size, equation stiffness, and method choice. Runge-Kutta 4th order typically has local error O(h⁵) and global error O(h⁴). Discontinuous or highly oscillatory functions may yield poor results.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

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