Differential Equations An Introduction for Engineers

As addressed in Chapter 1, we have a lot of different techniques for solving first order equations. However, not all differential equations are first order. A lot of physical systems in the world operate using higher order equations, particularly second order. Consider the system of a mass hanging from a spring. Newton’s second law tells us that the net force on the object equals the mass of the object times its acceleration. However, Hooke’s law for springs says that the force the spring exerts on the object is proportional to the distance this object is from the equilibrium position. Therefore, we get a relation between the acceleration of the object and the position. Since the acceleration is the second derivative (in time) of the position of the object, this naturally gives rise to a second order equation.

This means that we want to see what we can do with higher order equations as well. If we can manage to find solutions to these equations as well, then we can address more types of physical problems as well. However, increasing the order of the equation makes it significantly more difficult to find solutions. Even for linear equations, where in first order, we had an explicit method and formula for solutions, we need to put many more restrictions on higher order linear equations in order to have a direct method to generate solutions.