# The mathematical equation: y'' + 0.3*y' + y = 0
# Must be manually converted to a system:
def system(t, z):
    y, dy_dt = z
    d2y_dt2 = -0.3*dy_dt - y
    return [dy_dt, d2y_dt2]

# Then solve with scipy.integrate.solve_ivp
sol = solve_ivp(system, [0, 10], [1.0, 0.0])
# Which variable is which? What order? Error-prone! 😵‍💫

from odecast import var, Eq, solve

y = var("y")
eq = Eq(y.d(2) + 0.3*y.d() + y, 0)  # Exactly as written in textbooks!
sol = solve(eq, ivp={y: 1.0, y.d(): 0.0}, tspan=(0, 10))

# Crystal clear, impossible to mess up! 🎯

from odecast import var, Eq, solve

# Write the equation exactly as it appears in your textbook
y = var("y")
equation = Eq(y.d(2) + 0.3*y.d() + y, 0)

# Solve it (automatically converts to first-order system)
solution = solve(
    equation, 
    ivp={y: 1.0, y.d(): 0.0},  # y(0)=1, y'(0)=0
    tspan=(0, 10)
)

# Get results
import matplotlib.pyplot as plt
plt.plot(solution.t, solution[y])
plt.show()  # Beautiful decay curve! 📈

# Traditional nightmare:
def system(t, z): return [z[1], -0.3*z[1] - z[0]]

# Odecast elegance:
Eq(y.d(2) + 0.3*y.d() + y, 0)

# 2D harmonic oscillator in one line:
u = var("u", shape=2)
Eq(u.d(2) + u, 0)  # Automatically expands to u₀'' + u₀ = 0, u₁'' + u₁ = 0

# Clear error messages when you mess up:
y = var("y")
solve(Eq(y.d(2) + y, 0), ivp={y: 1.0})  # Missing y'(0)!
# ❌ ODEValidationError: Missing initial condition for y.d()

θ = var("θ")  # Angle
eq = Eq(θ.d(2) + 0.5*θ.d() + np.sin(θ), 0)  # Non-linear!
sol = solve(eq, ivp={θ: np.pi/4, θ.d(): 0}, tspan=(0, 20))

x, y = var("x"), var("y")  # Rabbits, foxes
eqs = [
    Eq(x.d() - x*(1 - y), 0),      # Rabbit growth
    Eq(y.d() - y*(-1 + x), 0)      # Fox dynamics  
]
sol = solve(eqs, ivp={x: 1, y: 1}, tspan=(0, 15))

u = var("u", shape=2)  # 2D position
eq = Eq(u.d(2) + 0.1*u.d() + u, 0)  # Damped 2D harmonic oscillator
sol = solve(eq, ivp={u: [1, 0], u.d(): [0, 1]}, tspan=(0, 10))
plt.plot(sol[u[0]], sol[u[1]])  # Phase space plot! 🌀

from odecast import t, var, Eq, solve

# Define a variable
y = var("y")

# Create an equation: y'' + 0.3*y' + y = 0 (damped harmonic oscillator)
eq = Eq(y.d(2) + 0.3*y.d() + y, 0)

# Solve with initial conditions y(0) = 1, y'(0) = 0
sol = solve(eq, ivp={y: 1.0, y.d(): 0.0}, tspan=(0.0, 10.0), backend="scipy")

# Access solution data
y_values = sol[y]        # Position over time
velocity = sol[y.d()]    # Velocity over time
times = sol.t            # Time points

# Evaluate at specific times
position_at_5s = sol.eval(y, 5.0)

---

*Built for scientists, engineers, and students who want to focus on the math, not the code.*

## Examples

The `examples/` directory contains comprehensive examples:

- `01_ivp_damped_oscillator.py` - Numeric IVP solving with visualization
- `02_symbolic_simple.py` - Symbolic solutions using SymPy backend  
- `03_mixed_orders.py` - Coupled systems with mixed derivative orders
- `04_vector_harmonic_oscillator.py` - 2D harmonic oscillator using vector variables
- `05_vector_mixed_system.py` - Mixed vector/scalar systems
- `06_vector_simple.py` - Simple vector variable introduction

Run any example:
```bash
python examples/01_ivp_damped_oscillator.py

# Solve numerically over a time span
sol = solve(eq, ivp=conditions, tspan=(0, 10), backend="scipy")

# Get exact symbolic solution
sol = solve(eq, backend="sympy")
expr = sol.as_expr(y)  # Returns SymPy expression

# Try numeric first, fall back to symbolic
sol = solve(eq, ivp=conditions, tspan=(0, 10), backend="auto")