Why are differential equations difficult to solve exactly?

Differential equations are fundamental to science and engineering because they describe how systems change over time, yet they are notoriously difficult to solve exactly. The primary reason is that equations capable of being solved through explicit formulas—using standard techniques from algebra and calculus—represent only a small minority of all possible differential equations.1

In most cases, it is impossible to produce an explicit formula for the unknown function. While mathematicians have developed specific techniques for certain classes of equations, such as linear or "almost" linear equations, many complex systems do not fall into these categories. When an exact solution cannot be found, researchers must rely on indirect methods or prove the existence of a solution without being able to construct it explicitly.231

Because of these limitations, numerical analysis is essential in practice. By using computers to approximate derivatives or replace complex functions with simpler ones (such as polynomials), scientists can obtain useful approximate solutions for systems that remain analytically intractable.14