A differential equation is a relation between a function and its derivatives. Solving it means finding all the functions that satisfy it and expressing them in the "simplest" form. One intuitive way to approach such equations is to start with familiar functions and ask which differential equations they solve.
Starting with the exponential function
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A fundamental function in mathematics, leading to one of the most elementary differential equations, is the exponential function, written f (x) = *ex. It is immediately clear that f ’(x) = ex = f (x). It therefore satisfies the equation f ’= f identically, or equivalently f ’– f = 0 (in practice, the variable x* is generally omitted from equations to make the notation more compact).
This makes it one of the "simplest" differential equations: it is linear, first-order (only the first derivative appears), has constant coefficients, and has a zero right-hand side.
Now let g (x) = x2. This new function g satisfies g’(x) = 2x.
