In ordinary equations the unknown is a number. But in a differential equation the unknown is a function, and the derivatives of the function occur in the equation.
The theory of differential equations was initiated by Isaac Newton (1643–1727) when he developed the theory in order to understand the motion of the planets. The theory has undergone tremendous evolution since then, and today all the laws of nature are expressed in terms of differential equations. Examples include the theory of gravitation, electromagnetism, fluid flow, and quantum mechanics. This means that the flow of blood in your arteries and veiins, the movemements of air and water, the casting of metals, and the flow of currents in your computer are all governed by differential equations.
Given the great varieties of phenomena described by differential equations, it may come as no surprise that their solution is in general not expressible by formulas.
Important research topics within the field include:
Do the equations have any solution at all, and is it unique?
How stable are the solutions?
How can you compute the solutions with sufficient accuracy?
