This chapter describes several techniques that can be brought to bear in order to solve initial-value problems as they occur in physics and engineering, such as the vibration of a mass-spring dashpot system. It considers the methods, which can be used to solve the class of problems that are known as initial-value problems. The chapter illustrates the Taylor series, the Runge–Kutta method, the Euler forward-integration method, the modified Euler method, and the Milne method. The techniques are but a small portion of the algorithms that are available for the integration of differential equations that are subject to initial conditions. The chapter discusses students’ use of these techniques. It explores the readers to the more relevant solution techniques that are used to solve initial-value differential equations. The chapter provides readers to select the solution process that is most efficient for their particular application.