Competencies: Mathematics > Differential Equations
Differential Equations
Body of Knowledge
| Topic | Description | Relevance | Career Tracks |
|---|---|---|---|
First-Order ODEs (Separable) |
Separation of variables, direct integration, initial value problems, existence and uniqueness (Picard-Lindelof theorem). |
High |
Engineer, Physicist, Data Scientist |
First-Order ODEs (Linear) |
Integrating factor method, standard form y' + P(x)y = Q(x), variation of parameters for first order. |
High |
Engineer, Physicist, Data Scientist |
First-Order ODEs (Exact) |
Exact equations M dx + N dy = 0, exactness condition dM/dy = dN/dx, integrating factors to make equations exact. |
Medium |
Engineer, Physicist |
Second-Order Linear ODEs |
Constant coefficient equations, characteristic equation, real/complex/repeated roots, general solutions, Wronskian and linear independence. |
Critical |
Engineer, Physicist, Data Scientist |
Laplace Transforms |
Definition, transform pairs, linearity, shifting theorems, transforms of derivatives, inverse transforms, solving IVPs via Laplace. |
High |
Engineer, Physicist, Data Scientist |
Systems of ODEs |
Matrix formulation, eigenvalue method, phase portraits, fundamental matrix, matrix exponential. |
High |
Engineer, Physicist, Data Scientist |
Stability & Phase Plane |
Equilibrium points, linearization, classification (node, saddle, spiral, center), Lyapunov stability, bifurcations. |
High |
Engineer, Physicist, Data Scientist |
Power Series Solutions |
Ordinary points, regular singular points, Frobenius method, recurrence relations, radius of convergence. |
Medium |
Physicist, Engineer |
Fourier Series |
Periodic functions, trigonometric series, Fourier coefficients, convergence (pointwise, uniform), Parseval’s theorem, applications to PDEs. |
High |
Engineer, Physicist, Data Scientist |
PDE Introduction (Heat, Wave, Laplace) |
Classification (elliptic, parabolic, hyperbolic), separation of variables, boundary conditions, heat equation, wave equation, Laplace equation. |
High |
Engineer, Physicist, Data Scientist |
Numerical Methods |
Euler’s method, improved Euler, Runge-Kutta (RK4), error analysis, stability of numerical schemes, finite difference methods. |
High |
Engineer, Data Scientist, Software Engineer |
Personal Status
To be populated after initial study and self-assessment.