First order differential equation on R of the form y’(x) = f(x,y(x)), Equivalent integral equation, Existence of approximate solutions of equation upto error $\epsilon$ by Cauchy-Euler method, Existence and uniqueness of solutions when $f$ is Lipshitz continuous in the second variable, Necessary conditions for f(x,y) to be Lipshitz continuous in y,Picard’s method of solutions of equation, Higher order differential equations, Vector valued ordinar differential equations, Reformulation of higher order differential equations as first order vector valued differential equations, Linear vector valued first order differential equation, Y’(x) = A Y(x) + C(x) — Homogeneous case, C =0, Characteristic values, characteristic vectors of square matrices, Solution when A is independent of x, Linear independence of solutions associated to characteristic values, General solution of the inhomogeneous equation, Peano’s approximation method for existence of solution.