Chapter 4: Linear differential equations

This chapter presents methods for the resolution of linear differential equations and special functions defined by second order linear differential equations. The solutions of the homogeneous equations with constant coefficients are defined by boundary conditions, otherwise the solutions are generally not unique. The other methods use the expansion in an orthonormal basis for inhomogeneous differential equations and reparametrizations. We present equivalences between linear and nonlinear second order equations and between classes of first order nonlinear equations such as Bernoulli's and Riccati's equations.