Differential equations

DIFFERENTIAL EQUATIONS 24 lectures, Michaelmas Term

Basic calculus

Informal treatment of diﬀerentiation as a limit, the chain rule, Leibnitz’s rule, Taylor series, informal treatment of O and o notation and l’Hˆopital’s rule; integration as an area, fundamental theorem of calculus, integration by substitution and parts. [3]
Informal treatment of partial derivatives, geometrical interpretation, statement (only) of symmetry of mixed partial derivatives, chain rule, implicit diﬀerentiation. Informal treatment of diﬀerentials, including exact diﬀerentials. Diﬀerentiation of an integral with respect to a parameter. [2]

Equations with constant coeﬃcients: exponential growth, comparison with discrete equations, series solution; modelling examples including radioactive decay. Equations with non-constant coeﬃcients: solution by integrating factor. [2]
Nonlinear ﬁrst-order equations Separable equations. Exact equations. Sketching solution trajectories. Equilibrium solutions, stability by perturbation; examples, including logistic equation and chemical kinetics. Discrete equations: equilibrium solutions, stability; examples including the logistic map. [4]

Complementary function and particular integral, linear independence, Wronskian (for second-order equations), Abel’s theorem. Equations with constant coeﬃcients and examples including radioactive sequences, comparison in simple cases with diﬀerence equations, reduction of order, resonance, transients, damping. Homogeneous equations. Response to step and impulse function inputs; introduction to the notions of the Heaviside step-function and the Dirac delta-function. Series solutions including statement only of the need for the logarithmic solution. [8]
Multivariate functions: applications Directional derivatives and the gradient vector. Statement of Taylor series for functions on Rn. Local extrema of real functions, classiﬁcation using the Hessian matrix. Coupled ﬁrst order systems: equivalence to single higher order equations; solution by matrix methods. Non-degenerate phase portraits local to equilibrium points; stability. Simple examples of ﬁrst- and second-order partial diﬀerential equations, solution of the wave equation in the form f(x + ct) + g(x−ct). [5]

J. Robinson An introduction to Diﬀerential Equations. Cambridge University Press, 2004
W.E. Boyce and R.C. DiPrima Elementary Diﬀerential Equations and Boundary-Value Problems (and associated web site: google Boyce DiPrima).(pdf)
Wiley, 2004 G.F.Simmons Diﬀerential Equations (with applications and historical notes). McGraw-Hill 1991
D.G. Zill and M.R. Cullen Diﬀerential Equations with Boundary Value Problems. Brooks/Cole 2001(pdf)