First Order Odes
When to Use
Use this skill when working on first-order-odes problems in odes pdes.
Decision Tree
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?
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} |
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
Verify Solution
- Substitute back into ODE
- Check initial/boundary conditions
sympy_compute.py dsolve "y' + y = x" --ics "{y(0): 1}"
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
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
uv run python -m runtime.harness scripts/sympy_compute.py dsolve "Derivative(y,x) + y" --ics "{y(0): 1}"
Z3_Equilibrium
uv run python -m runtime.harness scripts/z3_solve.py solve "f(y_star) == 0"
Key Techniques
From indexed textbooks:
- [Elementary Differential Equations and... (Z-Library)] Solving ODEs with MATLAB (New York: Cambridge REFERENCES cyan black NJ: Prentice-Hall, 1971). Mattheij, Robert, and Molenaar, Jaap, Ordinary Differential Equations in Theory and Practice Shampine, Lawrence F. Numerical Solution of Ordinary Differential Equations (New York: Chapman and Shampine, L.
- [Elementary Differential Equations and... (Z-Library)] Differential Equations: An Introduction to Modern Methods and Applications (2nd ed. Use the Laplace transform to solve the system 2e−t 3t α1 α2 , where α1 and α2 are arbitrary. How must α1 and α2 be chosen so that the solution is identical to Eq.
- [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)] Show that the rst order Adams–Bashforth method is the Euler method and that the rst order Adams–Moulton method is the backward Euler method. Show that the third order Adams–Moulton formula is yn+1 = yn + (h/12)(5fn+1 + 8fn − fn−1). Derive the second order backward differentiation formula given by Eq.
- [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.
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.
1---2name: first-order-odes3description: First Order Odes4---56# First Order Odes78## When to Use910Use this skill when working on first-order-odes problems in odes pdes.1112## Decision Tree1314151. **Classify the ODE**16 - Linear: y' + P(x)y = Q(x)?17 - Separable: y' = f(x)g(y)?18 - Exact: M(x,y)dx + N(x,y)dy = 0 with dM/dy = dN/dx?19 - Bernoulli: y' + P(x)y = Q(x)y^n?20212. **Select Solution Method**22 | Type | Method |23 |------|--------|24 | Separable | Separate and integrate |25 | Linear | Integrating factor e^{int P dx} |26 | Exact | Find potential function |27 | Bernoulli | Substitute v = y^{1-n} |28293. **Numerical Solution (IVP)**30 - `scipy.integrate.solve_ivp(f, [t0, tf], y0, method='RK45')`31 - For stiff systems: `method='Radau'` or `method='BDF'`32 - Adaptive step size: specify rtol/atol, not step size33344. **Verify Solution**35 - Substitute back into ODE36 - Check initial/boundary conditions37 - `sympy_compute.py dsolve "y' + y = x" --ics "{y(0): 1}"`38395. **Phase Portrait (Autonomous)**40 - Find equilibria: f(y*) = 041 - Analyze stability: sign of f'(y*)42 - `z3_solve.py solve "dy/dt == 0"`434445## Tool Commands4647### Scipy_Solve_Ivp48```bash49uv 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])"50```5152### Sympy_Dsolve53```bash54uv run python -m runtime.harness scripts/sympy_compute.py dsolve "Derivative(y,x) + y" --ics "{y(0): 1}"55```5657### Z3_Equilibrium58```bash59uv run python -m runtime.harness scripts/z3_solve.py solve "f(y_star) == 0"60```6162## Key Techniques6364*From indexed textbooks:*6566- [Elementary Differential Equations and... (Z-Library)] Solving ODEs with MATLAB (New York: Cambridge REFERENCES cyan black NJ: Prentice-Hall, 1971). Mattheij, Robert, and Molenaar, Jaap, Ordinary Differential Equations in Theory and Practice Shampine, Lawrence F. Numerical Solution of Ordinary Differential Equations (New York: Chapman and Shampine, L.67- [Elementary Differential Equations and... (Z-Library)] Differential Equations: An Introduction to Modern Methods and Applications (2nd ed. Use the Laplace transform to solve the system 2e−t 3t α1 α2 , where α1 and α2 are arbitrary. How must α1 and α2 be chosen so that the solution is identical to Eq.68- [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.69- [Elementary Differential Equations and... (Z-Library)] Show that the rst order Adams–Bashforth method is the Euler method and that the rst order Adams–Moulton method is the backward Euler method. Show that the third order Adams–Moulton formula is yn+1 = yn + (h/12)(5fn+1 + 8fn − fn−1). Derive the second order backward differentiation formula given by Eq.70- [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, 61-71.7172## Cognitive Tools Reference7374See `.claude/skills/math-mode/SKILL.md` for full tool documentation.
DojoGenesis