By name: differential-equations description: Equations involving derivatives license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics
What I do
- Solve first-order ODEs (separable, linear, exact)
- Solve higher-order linear ODEs with constant coefficients
- Apply Laplace transforms for ODE solutions
- Solve systems of differential equations
- Analyze stability of equilibrium solutions
- Solve partial differential equations
When to use me
When modeling dynamic systems, physics problems, or any system involving rates of change.
Key Concepts
- First-Order Linear: dy/dx + P(x)y = Q(x), integrate factor μ = e^{∫Pdx}
- Separable: dy/dx = f(x)g(y), ∫dy/g(y) = ∫f(x)dx
- Second-Order Linear: ay'' + by' + cy = 0, characteristic r² + br/a + c/a = 0
- Laplace Transform: L{f(t)} = ∫₀^∞ e^{-st}f(t)dt, converts ODEs to algebraic
- Particular Solution: Sum of homogeneous + particular solves full equation
- Phase Lines: Visualize equilibrium stability for autonomous ODEs