Picard's Theorem Calculator

Calculate Result Using Picard's Theorem





Formula

The iterative method used according to Picard's Theorem is:

\[ y_{n+1} = y_0 + \int_{t_0}^{t} f(s, y_n(s)) \, ds \]

Where:

What is Picard's Theorem?

Picard's Theorem is a fundamental result in the theory of ordinary differential equations. It states that if a function satisfies certain conditions, then there exists a unique solution to the differential equation in the neighborhood of a given point. The theorem provides a method for constructing a sequence of approximate solutions that converge to the actual solution.

Example Calculation

Let's assume the following values:

Using the iterative method, the result after 10 iterations is:

\[ y_{10} = y_0 + \int_{t_0}^{t} f(s, y_9(s)) \, ds = 1 + \int_{0}^{1} s \cdot y_9(s) \, ds \]

Note: The actual value of the integral and function \(f(s, y)\) need to be defined for a specific differential equation.