first-order-odes

# First Order Odes ## When to Use Use this skill when working on first-order-odes problems in odes pdes. ## Decision Tree 1. **Classify the ODE** - Linear: y' + P(x)y = Q(x)? - Separable: y' = f(x)g(y)? - Exact: M(x,y)dx + N(x,y)dy = 0 with dM/dy = dN/dx? - Bernoulli: y' + P(x)y = Q(x)y^n? 2. **Select Solution Method** | Type | Method | |------|--------| | Separable | Separate and integrate | | Linear | Integrating factor e^{int P dx} | | Exact | Find potential function | | Bernoulli | Substitute v = y^{1-n} | 3. **Numerical Solution (IVP)** - `scipy.integrate.solve_ivp(f, [t0, tf], y0, method='RK45')` - For stiff systems: `method='Radau'` or `method='BDF'` - Adaptive step size: specify rtol/atol, not step size 4. **Verify Solution** - Substitute back into ODE - Check initial/boundary conditions - `sympy_compute.py dsolve "y' + y = x" --ics "{y(0): 1}"` 5. **Phase Portrait (Autonomous)** - Find equilibria: f(y*) = 0 - Analyze stability: sign of f'(y*) - `z3_solve.py solve "dy/dt == 0"` ## Tool Commands ### Scipy_Solve_Ivp ```bash uv run python -c "from scipy.integrate import solve_ivp; sol = solve_ivp(lambda t, y: -y, [0, 5], [1]); print('y(5) =', sol.y[0][-1])" ``` ### Sympy_Dsolve ```bash uv run python -m runtime.harness scripts/sympy_compute.py dsolve "Derivative(y,x) + y" --ics "{y(0): 1}" ``` ### Z3_Equilibrium ```bash uv run python -m runtime.harne...