Course Slides & Notes
MA1035 • 12 Sessions
Introduction to ODEs
Fundamental definitions, mathematical modeling principles, rate of growth/decay hypotheses, qualitative analysis, and slope fields.
Non-homogeneous 1st Order Linear ODEs
Standard linear form, integrating factor method, variation of parameters, and applications to dilution and thermal models.
Higher Order Linear Differential Equations with cc
Homogeneous linear differential equations with constant coefficients, characteristic auxiliary equations, real distinct, repeated, and complex roots.
Non-homogeneous Higher Order LODEs with cc
General solution structure y = yh + yp, method of undetermined coefficients, trial functions table, and polynomial/trigonometric forcing.
Modelling with Higher Order Constant-Coefficient LODEs
Engineering dynamic models: industrial heating processes, factory climate regulation, elevator suspension dynamics, and electric motor velocity.
Systems of LODEs
Coupled dynamic systems, multi-species predator-prey equations, state-space vector-matrix form X' = AX, and superposition principle.
Eigenfunctions for Systems of LODEs
Eigenvalues, eigenvectors, characteristic equations det(A - lambda I) = 0, fundamental set of solutions, and linear independence.
Generalized Eigenvectors & Complex Eigenvalues
Defective matrix systems, chains of generalized eigenvectors (A - lambda I)v2 = v1, complex conjugate eigenvalue solutions, and phase plane portraits.
The Laplace Transform
Integral transform concept, formal definition, Laplace transforms of elementary functions, algebraic linearity, and piecewise forcing functions.
Applied Laplace Transform & IVPs
Inverse Laplace transform, operational rules for derivatives L{y'} and L{y''}, partial fractions decomposition, and solving complete Initial Value Problems.
1st Translation Theorem (s-shift)
Shift in the s-domain L{e^(at) f(t)} = F(s - a), completing the square in denominators, inverse calculations, and damped oscillatory dynamic systems.
2nd Translation Theorem (t-shift)
Heaviside unit step function, time-delay translation L{f(t-a) U(t-a)} = e^(-as) F(s), piecewise switching inputs, and impulsive responses in engineering.