Differential Equations

Phase 3: Differential Equations & Transforms

Syllabus Goal

Linear differential equations and Laplace transforms. Focus on linear first-order and second-order ODEs with constant coefficients.

Ordinary Differential Equations (ODEs)

• First Order: $\frac{dy}{dx} + P(x)y = Q(x)$. Use integrating factor $I(x) = e^{\int P(x)dx}$.
• Second Order: $ay’’ + by’ + cy = 0$ (Homogeneous). Characteristic equation $ar^2 + br + c = 0$.

Laplace Transforms

$$ \mathcal{L}{f(t)} = F(s) = \int_0^\infty e^{-st} f(t) dt $$

• Used to convert ODEs into algebraic equations.
• Inversion via partial fractions or tables.

Implementation in Zig

• Numerical solvers (Euler, Runge-Kutta methods).
• Simulating damped harmonic oscillators.