Isaac Newton originally studies behaviour of dynamical systems using differential equations. Mathematical models are often described with differential equations and they are widely used in many fields, including physics, chemistry, biology, economics etc. Mathematical methods used in solving differential equation also play an important in advanced studies. Key topics to learn are: Wronskians, series solutions, reduction of order, variation of parameters, Frobenius normal form, Bessel's equation, Legendre's equation, two points boundary problem, Fredholm alternative, Green's function, complete orthogonal system, Sturm-Liouville problems (inhomogeneous, singular), basic partial differential equation, elliptic eigen-problems, heat equation and wave equation.
- Boyce W.E. and DiPrima R.C. Elementary Differential Equations and Boundary Value Problems
- Dawkins P. Differential Equations (FREE!)
- Forsyth A.R. A Treatise on Differential Equations
- Ikenaga B. Differential Equations Notes (FREE!)
- Shapiro B.E. Lecture Notes on Differential Equations (FREE!)
- Simmons G.F. and Krantz S.G. Differential Equations: Theory, Technique, and Practice - Both Simmons and Krantz are writers for mathematicians.
- Veeh J.A. Lectures Notes on Ordinary Differential Equations [.pdf] (FREE!)