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 = c1e^{r1x} + c2e^{r2x}
   • Repeated root: y = (c1 + c2x)e^{rx}
   • Complex roots a +/- bi: y = e^{ax}(c1cos(bx) + c2sin(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 = u1y1 + u2y2
   • 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

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

uv run python -m runtime.harness scripts/sympy_compute.py solve "r**2 + r + 1" --var r

Sympy_Dsolve_2Nd

uv run python -m runtime.harness scripts/sympy_compute.py dsolve "Derivative(y,x,2) + y"

Key Techniques

From indexed textbooks:

• [An Introduction to Numerical Analysis... (Z-Library)] Modern Numerical Methods for Ordinary Wiley, New York. User's guide for DVERK: A subroutine for solving non-stiff ODEs. Keller (1966), Analysis of Numerical Methods.
• [Elementary Differential Equations and... (Z-Library)] Riccati equation and that y1(t) = 1 is one solution. Use the transformation suggested in Problem 33, and nd the linear equation satised by v(t). Find v(t) in the case that x(t) = at, where a is a constant.
• [An Introduction to Numerical Analysis... (Z-Library)] Test results on initial value methods for non-stiff ordinary differential equations, SIAM J. Comparing numerical methods for Fehlberg, E. Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnumg mit Schrittweiten-Kontrolle und ihre Anwendung auf Warme leitungsprobleme, Computing 6, 61-71.
• [Elementary Differential Equations and... (Z-Library)] Two papers by Robert May cited in the text are R. May,“Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos,” Science 186 (1974), pp. Biological Populations Obeying Difference Equations: Stable Points, Stable Cycles, and Chaos,” Journal of Theoretical Biology 51 (1975), pp.
• [An Introduction to Numerical Analysis... (Z-Library)] COLSYS: collocation software for boundary-value ODEs, ACM Trans. Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations. Elementary Differential Equations and Boundary Value Problems, 4th ed.