Taylor Series ODE Solver Calculator

Calculator Input Panel

ODE expression f(x, y)

Examples: x + y, sin(x) - y, y*(1-y/10)

Initial x₀

Initial y₀

Target x

Step size h

Taylor order

Derivative epsilon

Smaller values increase sensitivity. Typical range: 1e-5 to 1e-3.

Output decimals

Supported functions

sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log, log10, sqrt, abs, floor, ceil, pow, pi

Example Data Table

Sample setup: y' = x + y, x0 = 0, y0 = 1, target x = 0.5, h = 0.1, order = 4.

Example data could not be generated.

Formula Used

Taylor update:

y(x+h) ≈ y(x) + h·y'(x) + h²/2!·y''(x) + h³/3!·y'''(x) + ... + hⁿ/n!·y⁽ⁿ⁾(x)

For a first-order initial value problem y' = f(x, y), the calculator builds each step from the local Taylor polynomial. The first derivative is y' = f(x, y).

Higher derivatives are obtained with the total derivative operator: Dg = ∂g/∂x + f(x, y)·∂g/∂y. Therefore, y'' = D(f), y''' = D(y''), and so on.

Because users can enter a general expression, the calculator estimates the needed partial derivatives numerically with central differences. That makes the tool flexible for many first-order ODE models without requiring symbolic algebra input.

How to Use This Calculator

Enter the ODE as f(x, y). Use explicit multiplication such as 2*x and x*y.
Provide the initial condition (x0, y0).
Set the target x-value and choose the step size h.
Select the Taylor order. Higher order can improve accuracy but may increase sensitivity.
Choose a derivative epsilon for numerical differentiation. Start with 0.0001 if unsure.
Press Solve ODE to generate the result above the form.
Review the plotted solution, step table, and local estimate before exporting CSV or PDF.

Frequently Asked Questions

1) What type of equations does this solver handle?

It handles first-order initial value problems written as y' = f(x, y). You provide the function, start point, step size, and target x-value.

2) Why is Taylor order important?

The order decides how many derivative terms are included in each local polynomial. Higher order often improves accuracy, especially for smooth functions and moderate steps.

3) What does derivative epsilon control?

It controls the central-difference spacing used to estimate higher derivatives. If it is too large, derivatives become coarse. If too small, roundoff noise may grow.

4) Why can smaller h improve results?

A smaller step keeps each Taylor expansion closer to its base point. That usually reduces truncation error and improves stability for rapidly changing solutions.

5) What is the estimate column showing?

It shows the difference between the chosen order step and the one-order-lower step. It is a practical local indicator, not a guaranteed global error bound.

6) Can this replace symbolic differentiation?

No. It is a flexible numerical approach. Symbolic Taylor methods can be more exact for known equations, but they require analytic derivative expressions.

7) Which expressions should I avoid?

Avoid undefined regions, discontinuities, and expressions that explode quickly. Stiff or highly singular equations may require very small steps or different numerical methods.

8) What functions can I use in the input?

You can use x, y, parentheses, powers, and common functions such as sin, cos, tan, exp, log, sqrt, abs, and related inverse or hyperbolic functions.