basic_examples.py

"""
Basic examples of the Puiseux differential algebra pipeline.

Run with:
    python examples/basic_examples.py

Requires:
    - sympy
    - DifferentialAlgebra (compiled C extension)
"""

from sympy import var
from DifferentialAlgebra import indexedbase

# Setup shared symbols
xi = var("xi")
x, y = indexedbase("x,y")

SEP = "=" * 70


def example_1_dy2_minus_x():
    """ẏ² - x = 0  →  y = C + (2/3)x^{3/2}"""
    print(f"\n{SEP}")
    print("Example 1: ẏ² - x = 0")
    print(f"Expected: y = z₀ + (2/3)x^{3/2}")
    print(SEP)

    from src.puiseux_pipeline import find_puiseux_solutions, filter_nontrivial

    P = y[xi] ** 2 - x
    sols = find_puiseux_solutions(P, b_max=10, verbose=True)
    sols = filter_nontrivial(sols)

    print(f"\nNon-trivial solutions found: {len(sols)}")
    for s in sols:
        print(f"  (a,b) = ({s['a']}, {s['b']})")
        print(f"  FPS series : {s['series']}")
        print(f"  Puiseux    : {s['puiseux_solution']}")


def example_2_dy3_minus_x2():
    """ẏ³ - x² = 0  →  y = C + (3/5)x^{5/3}"""
    print(f"\n{SEP}")
    print("Example 2: ẏ³ - x² = 0")
    print(f"Expected: y = z₀ + (3/5)x^{5/3}")
    print(SEP)

    from src.puiseux_pipeline import find_puiseux_solutions, filter_nontrivial

    P = y[xi] ** 3 - x**2
    sols = find_puiseux_solutions(P, b_max=15, verbose=True)
    sols = filter_nontrivial(sols)

    print(f"\nNon-trivial solutions found: {len(sols)}")
    for s in sols:
        print(f"  (a,b) = ({s['a']}, {s['b']})")
        print(f"  Puiseux: {s['puiseux_solution']}")


def example_3_class_P_32_1():
    """8ẏ³ - 24ẏ² - 243y + 32 = 0 → y = 2x + 3x^{3/2}"""
    print(f"\n{SEP}")
    print("Example 3: Class P_{(3/2,1)}")
    print("8ẏ³ - 24ẏ² - 243y + 32 = 0")
    print(f"Expected: y = 2x + 3x^{3/2}")
    print(SEP)

    from src.puiseux_pipeline import find_puiseux_solutions, filter_nontrivial

    P = 8 * y[xi] ** 3 - 24 * y[xi] ** 2 - 243 * y + 32
    sols = find_puiseux_solutions(P, b_max=10, verbose=True)
    sols = filter_nontrivial(sols)

    print(f"\nNon-trivial solutions found: {len(sols)}")
    for s in sols:
        print(f"  (a,b) = ({s['a']}, {s['b']})")
        if s.get("puiseux_solution") is not None:
            print(f"  Puiseux: {s['puiseux_solution']}")


def example_4_fast_decision():
    """Demonstrate the FAST DECISION algorithm."""
    print(f"\n{SEP}")
    print("Example 4: Fast Decision Algorithm")
    print(SEP)

    from src.two_algorithms import fast_decision

    test_cases = [
        ("ẏ² - x", y[xi] ** 2 - x),
        ("ẏ³ - x²", y[xi] ** 3 - x**2),
        ("8ẏ³ - 24ẏ² - 243y + 32",
         8 * y[xi] ** 3 - 24 * y[xi] ** 2 - 243 * y + 32),
        ("ẏ³ - x", y[xi] ** 3 - x),  # should be NO
    ]

    for name, P in test_cases:
        result, reason = fast_decision(P, verbose=False)
        print(f"\n  {name}:")
        print(f"    Decision: {result}")
        print(f"    Reason:   {reason}")


def example_5_second_order():
    """ÿ² - y - x² - 1 = 0 (second-order AODE)"""
    print(f"\n{SEP}")
    print("Example 5: Second-order AODE")
    print("ÿ² - y - x² - 1 = 0")
    print(SEP)

    from src.puiseux_pipeline import find_puiseux_solutions, filter_nontrivial

    P = y[xi, xi] ** 2 - y - x**2 - 1
    sols = find_puiseux_solutions(P, b_max=8, verbose=True)
    sols = filter_nontrivial(sols)

    print(f"\nNon-trivial solutions found: {len(sols)}")
    for s in sols:
        print(f"  (a,b) = ({s['a']}, {s['b']})")
        if s.get("puiseux_solution") is not None:
            print(f"  Puiseux: {s['puiseux_solution']}")


if __name__ == "__main__":
    print("Puiseux Differential Algebra — Examples")
    print("=" * 70)
    print("Make sure DifferentialAlgebra is installed:")
    print("  cd DifferentialAlgebra/bmi-install-sympy-c99/packaging")
    print("  python3 setup.py build && pip install -e .")
    print()

    try:
        example_1_dy2_minus_x()
    except Exception as e:
        print(f"Example 1 failed: {e}")

    try:
        example_2_dy3_minus_x2()
    except Exception as e:
        print(f"Example 2 failed: {e}")

    try:
        example_3_class_P_32_1()
    except Exception as e:
        print(f"Example 3 failed: {e}")

    try:
        example_4_fast_decision()
    except Exception as e:
        print(f"Example 4 failed: {e}")

    try:
        example_5_second_order()
    except Exception as e:
        print(f"Example 5 failed: {e}")

    print(f"\n{SEP}")
    print("Done.")