second-order-odes

# Second Order Odes ## When to Use Use this skill when working on second-order-odes problems in odes pdes. ## Decision Tree 1. **Classify the ODE** - Constant coefficients: ay'' + by' + cy = f(x)? - Variable coefficients: y'' + P(x)y' + Q(x)y = R(x)? - Cauchy-Euler: x^2 y'' + bxy' + cy = 0? 2. **Homogeneous with Constant Coefficients** - Characteristic equation: ar^2 + br + c = 0 - Distinct real roots: y = c1*e^{r1*x} + c2*e^{r2*x} - Repeated root: y = (c1 + c2*x)e^{r*x} - Complex roots a +/- bi: y = e^{ax}(c1*cos(bx) + c2*sin(bx)) - `sympy_compute.py solve "a*r**2 + b*r + c" --var r` 3. **Particular Solution (Non-homogeneous)** - Undetermined coefficients: guess based on f(x) - Variation of parameters: y_p = u1*y1 + u2*y2 - `sympy_compute.py dsolve "y'' + y = sin(x)"` 4. **Numerical Solution** - Convert to first-order system: let v = y', then v' = y'' - `solve_ivp(system, [t0, tf], [y0, v0])` 5. **Boundary Value Problems** - Shooting method: guess initial slope, iterate - `scipy.integrate.solve_bvp(ode, bc, x, y_init)` ## Tool Commands ### Scipy_Solve_Ivp_System ```bash uv run python -c "from scipy.integrate import solve_ivp; sol = solve_ivp(lambda t, Y: [Y[1], -Y[0]], [0, 10], [1, 0]); print('y(10) =', sol.y[0][-1])" ``` ### Sympy_Charpoly ```bash uv run python -m runtime.harness scripts/sympy_compute.py solve "r**2 + r + 1" --var r ``` ### Sympy_Dsolve_2Nd ```bash uv...