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CHAPTER 2 First-Order Di↵erential Equations 1. Introduction: Motion of a Falling Body Problem. An object falls through the air toward earth. Assuming that the only forces acting on the object are gravity and air resistance, determine the velocity of the object as a function of time. With F the total force on the object, m the mass and v the velocity of the dv object, gives the acceleartion of the object. By Newton’s second law, dt dv m = F. dt Here we will assume v is positive when it is directed downward. Also, near the Earth’s surface, the force due to gravity is mg where g is the acceleration due to gravity. Air resistance, which is proportional to velocity, is given by bv where b is a positive constant depending on the density of the air and the shape of the object. The negative sign is since the air resistance acts opposite to gravity. Thus we have the first order DE dv m = mg bv. dt We solve this equation using separation of variables, treating dv and dt as differentials. Assuming m 6= 0 and mg bv 6= 0, we can get dv dt = mg bv m 17