Engineering Mathematics III: Differential Equations & Transforms

Modelling change over time. Ordinary differential equations from fundamentals to systems: first-order methods, higher-order linear equations and vibrations, systems and state space, power-series and special equations, Laplace transform methods, Fourier series and the Fourier transform.

• Differential Equation Fundamentals — Differential equations relate a function to its derivatives — the language of dynamic systems. Learn ODE vs PDE, order and degree, linear vs non-linear, initial and boundary conditions, general vs particular solutions, existence and uniqueness, direction fields, and the modelling workflow from physical law to equation.
• First-Order ODEs — The workhorse equations of engineering. Learn separable equations, homogeneous (y/x) equations, exact equations and integrating factors, linear first-order equations with the integrating factor e^∫P, Bernoulli equations, orthogonal trajectories, and applications: RC/RL circuits, Newton's cooling, mixing tanks, population growth and motion with drag — with an RK4 check.
• Higher-Order Linear ODEs — Second-order equations model vibrations and RLC circuits. Learn homogeneous constant-coefficient equations and the auxiliary (characteristic) equation, real, repeated and complex roots, damping regimes, non-homogeneous equations with complementary function + particular integral, undetermined coefficients, variation of parameters, and resonance.
• Systems of ODEs & State Space — Coupled equations describe interacting parts: two-tank systems, predator–prey, multi-mass vibrations, electrical networks. Learn elimination, matrix form x′ = Ax, eigenvalue solutions, converting higher-order ODEs to first-order systems, state-space models and stability from eigenvalues.
• Series Solutions & Taylor Series — When closed-form solutions fail, power series succeed. Learn Taylor and Maclaurin series, radius of convergence, power-series solutions of ODEs, ordinary and singular points, recurrence relations, and the Frobenius method for regular singular points.
• Special Differential Equations — A handful of named ODEs recur throughout physics and engineering. Learn Legendre's equation and polynomials, Bessel's equation and functions, Hermite, Laguerre and Chebyshev equations, where each arises (spheres, cylinders, quantum oscillators, approximation), and key properties like orthogonality.
• Laplace Transform Methods — The Laplace transform converts linear ODEs with initial conditions into algebra. Learn the definition and existence, standard transform pairs, linearity, first and second shifting theorems, transforms of derivatives and integrals, unit-step and impulse functions, inverse transforms by partial fractions, convolution, and initial/final value theorems.
• Fourier Series — Any periodic function can be written as a sum of sines and cosines. Learn periodic functions, Euler formulas for the coefficients, even and odd symmetry (cosine and sine series), half-range expansions, convergence and the Gibbs phenomenon, Parseval's theorem, and the complex exponential form.
• Fourier Transform — Any signal can be built from sinusoids. Learn the Fourier series of periodic signals (square wave harmonics), the Fourier transform and its properties (linearity, time shift, scaling, convolution ↔ multiplication, Parseval), spectra of common signals, bandwidth, and frequency response of LTI systems.