Picard’S Theorem Calculator

Calculate exact or Picard iteration approximations of y(t) for first-order initial value problems from y′=f(t,y), t₀, y(t₀), and target t.

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Picard Iteration Formula

For an IVP y′ = f(t, y) with y(t₀) = y₀, Picard iteration constructs a sequence of functions converging to the unique solution on some interval around t₀, provided f is Lipschitz continuous in y.

y_{k+1}(t)=y_0+\int_{t_0}^{t} f\!\bigl(s,\,y_k(s)\bigr)\,ds

What is Picard’s Theorem?

The Picard-Lindelöf theorem (also called the Cauchy-Lipschitz theorem) states: if f(t, y) is continuous on a closed rectangle R centered at (t₀, y₀) and satisfies a Lipschitz condition in y with constant L, then the IVP y′ = f(t, y), y(t₀) = y₀ has exactly one solution on |t − t₀| ≤ h, where h = min(a, b/M) and M = sup|f| on R.

Named after Emile Picard (1856–1941), who formalized the iterative proof in 1890, and Ernst Lindelöf (1870–1946), who extended it to local Lipschitz conditions in 1894. Earlier foundational work by Cauchy (1820s) and Lipschitz (1876) yields the alternate name Cauchy-Lipschitz theorem.

Existence and Uniqueness Theorem Comparison

Lipschitz Constants for the Built-In Equations

When Uniqueness Fails

The Lipschitz condition is not merely technical. When it fails at a point, multiple distinct solutions can pass through the same initial condition, or solutions can escape to infinity in finite time.

Convergence of Picard Iterates

For y′ = y, y(0) = 1 (exact solution e^t), each Picard iterate equals a partial Taylor sum. The table below shows exact convergence values at t = 1.

For stiff equations or large intervals, Runge-Kutta methods reach comparable accuracy in far fewer function evaluations. Picard iteration is used primarily as a theoretical tool; Euler and RK4 are the practical choices for numerical computation.

How to Calculate Using Picard’s Theorem

1. Write the IVP in standard form: y′ = f(t, y) with y(t₀) = y₀.
2. Verify the Lipschitz condition: compute |∂f/∂y| on the target domain and confirm it is bounded above by some constant L.
3. Choose target point t and start with y₀(t) = y₀ as the constant initial guess.
4. Apply the Picard integral formula repeatedly. For numerics, increase integration steps first (per-iterate accuracy) before increasing the iterate count (convergence depth).
5. Stop when successive iterates differ by less than your error tolerance, then compare with the Exact Solution tab above.

Example Problem:

Equation: y′ = y + t, t₀ = 0, y(0) = 1, target t = 0.5. Lipschitz constant L = 1 (global). Exact solution: y(t) = 2e^t − t − 1, so y(0.5) ≈ 1.797443. Using the Picard Iteration tab with 5 iterations and 100 integration steps produces a result accurate to at least 4 decimal places.

Applications