This chapter introduces Picard’s method of successive approximations. This method is sometimes used to approximate the solution of the initial value problem \(\displaystyle y' = f(x,y), \quad y(x_0) = y_0 \qquad \hbox{(1)} \) for a given function f, assuming it has one, when (1) does not have a closed-form solution or it is difficult to find. A number of examples using Picard’s method are presented, such as calculating several Picard approximations of the solution of \(\displaystyle y' = 2\sin {x} - y, \quad y(0) = -1. \qquad \hbox{(2)} \) Eventually, it becomes clear from these approximations that the solution of (2) is \(y(x) = \sin {x} - \cos {x}\) , which is what the integrating factor method or the variation of parameters formula in Chap. 5 would yield. The real purpose of examples like (2) is to get students comfortable with Picard’s method before it is used to prove an existence and uniqueness theorem that guarantees a unique solution of (1) exists when the function f is continuous and satisfies a Lipschitz condition with respect to y on a closed rectangular region \(\mathcal {R}\) . In another version of this theorem, it is assumed that both f and the partial derivative \(\partial {f}/\partial {y}\) are continuous on \(\mathcal {R}\) . Another existence and uniqueness theorem is stated for initial value problems involving the second-order differential equation \(\displaystyle y'' = f(x, y, y'). \qquad \hbox{(3)} \) Examples are provided demonstrating how to use this theorem to establish the existence and uniqueness of the solution of (3) for a given function f and set of initial conditions. In one of these examples, the theorem is used to prove that \(\displaystyle y'' = xy, \quad y(0) = 0, \quad y'(0) = 1 \qquad \hbox{(4)} \) has a unique solution. Then Picard’s method is used to obtain this solution, which turns out to be a convergent series that cannot be expressed in terms of the elementary functions of calculus. The differential equation in (3), called Airy’s equation, is important in many fields, such as optics and fluid dynamics.