While there are many general techniques for analytically solving classes of ODEs, the only practical solution technique for complicated equations is to use numerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The most popular of these is the Runge-Kutta method , but many others have been developed, including the collocation method and Galerkin method . A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics.

Hence, in practice, we can safely treat $\diff{x}{t}$ like a fraction when used in this context of forming an integral to solve a differential equation. To solve the equation $\diff{x}{t}=ax+b$, we multiply both sides of the equation by $dt$ and divide both sides of the equation by $ax+b$ to get \begin{gather*} \frac{dx}{ax+b} = dt. \end{gather*} Then, we integrate both sides to obtain \begin{gather*} \int \frac{dx}{ax+b} = \int dt. \end{gather*} Just remember that these manipulations are really a shortcut way to denote using the chain rule.