[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6NVK2STR0285GSKN44DT2S0":3,"skill-same:01M6NVK2STR0285GSKN44DT2S0":55},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":18,"publishedAt":26,"linkOnly":27,"markdownContent":28,"markdownBody":29,"frontMatter":30,"refName":33,"commitSha":34,"skillDirectory":35,"forksCount":14,"repositoryHtmlUrl":36,"repositoryHomepageUrl":6,"ownerHtmlUrl":37,"ownerBlogUrl":38,"ownerDescription":39,"updatedAt":40,"install":41,"resources":54},"01M6NVK2STR0285GSKN44DT2S0","second-order-odes",null,"Problem-solving strategies for second order odes in odes pdes","rubicanjr\u002FFinCognis","rubicanjr","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F234912519?v=4","01M6NT156NND8J474G34SB3923","Rubi","01M6NT15B70RWXE1N3YV0BZY8A",0,"MIT","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Fodes-pdes\u002Fsecond-order-odes\u002FSKILL.md",[],[19,20,21,22,23,24,25],"second order odes","Bash","read","problem-solving","strategies","problem","for","2026-11-10 10:12:30",false,"---\nname: second-order-odes\ndescription: \"Problem-solving strategies for second order odes in odes pdes\"\nallowed-tools: [Bash, Read]\n---\n\n# Second Order Odes\n\n## When to Use\n\nUse this skill when working on second-order-odes problems in odes pdes.\n\n## Decision Tree\n\n\n1. **Classify the ODE**\n   - Constant coefficients: ay'' + by' + cy = f(x)?\n   - Variable coefficients: y'' + P(x)y' + Q(x)y = R(x)?\n   - Cauchy-Euler: x^2 y'' + bxy' + cy = 0?\n\n2. **Homogeneous with Constant Coefficients**\n   - Characteristic equation: ar^2 + br + c = 0\n   - Distinct real roots: y = c1*e^{r1*x} + c2*e^{r2*x}\n   - Repeated root: y = (c1 + c2*x)e^{r*x}\n   - Complex roots a +\u002F- bi: y = e^{ax}(c1*cos(bx) + c2*sin(bx))\n   - `sympy_compute.py solve \"a*r**2 + b*r + c\" --var r`\n\n3. **Particular Solution (Non-homogeneous)**\n   - Undetermined coefficients: guess based on f(x)\n   - Variation of parameters: y_p = u1*y1 + u2*y2\n   - `sympy_compute.py dsolve \"y'' + y = sin(x)\"`\n\n4. **Numerical Solution**\n   - Convert to first-order system: let v = y', then v' = y''\n   - `solve_ivp(system, [t0, tf], [y0, v0])`\n\n5. **Boundary Value Problems**\n   - Shooting method: guess initial slope, iterate\n   - `scipy.integrate.solve_bvp(ode, bc, x, y_init)`\n\n\n## Tool Commands\n\n### Scipy_Solve_Ivp_System\n```bash\nuv 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])\"\n```\n\n### Sympy_Charpoly\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py solve \"r**2 + r + 1\" --var r\n```\n\n### Sympy_Dsolve_2Nd\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py dsolve \"Derivative(y,x,2) + y\"\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [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.\n- [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.\n- [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.\n- [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.\n- [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.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n","# Second Order Odes\n\n## When to Use\n\nUse this skill when working on second-order-odes problems in odes pdes.\n\n## Decision Tree\n\n\n1. **Classify the ODE**\n   - Constant coefficients: ay'' + by' + cy = f(x)?\n   - Variable coefficients: y'' + P(x)y' + Q(x)y = R(x)?\n   - Cauchy-Euler: x^2 y'' + bxy' + cy = 0?\n\n2. **Homogeneous with Constant Coefficients**\n   - Characteristic equation: ar^2 + br + c = 0\n   - Distinct real roots: y = c1*e^{r1*x} + c2*e^{r2*x}\n   - Repeated root: y = (c1 + c2*x)e^{r*x}\n   - Complex roots a +\u002F- bi: y = e^{ax}(c1*cos(bx) + c2*sin(bx))\n   - `sympy_compute.py solve \"a*r**2 + b*r + c\" --var r`\n\n3. **Particular Solution (Non-homogeneous)**\n   - Undetermined coefficients: guess based on f(x)\n   - Variation of parameters: y_p = u1*y1 + u2*y2\n   - `sympy_compute.py dsolve \"y'' + y = sin(x)\"`\n\n4. **Numerical Solution**\n   - Convert to first-order system: let v = y', then v' = y''\n   - `solve_ivp(system, [t0, tf], [y0, v0])`\n\n5. **Boundary Value Problems**\n   - Shooting method: guess initial slope, iterate\n   - `scipy.integrate.solve_bvp(ode, bc, x, y_init)`\n\n\n## Tool Commands\n\n### Scipy_Solve_Ivp_System\n```bash\nuv 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])\"\n```\n\n### Sympy_Charpoly\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py solve \"r**2 + r + 1\" --var r\n```\n\n### Sympy_Dsolve_2Nd\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py dsolve \"Derivative(y,x,2) + y\"\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [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.\n- [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.\n- [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.\n- [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.\n- [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.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n",{"name":5,"description":7,"allowed-tools":31},[20,32],"Read","main","70e1c350500ee750987420bbae48837afa906b7e","skills\u002Fmath\u002Fodes-pdes\u002Fsecond-order-odes","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis","https:\u002F\u002Fgithub.com\u002Frubicanjr","linktr.ee\u002Ffinanskodu","Industrial Engineer & CIO @FinansKodu. 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