Solving by direct integration

Differential equations of the form
\[ \frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}(x) \]
can be solved by direct integration.

Here,
\[ y = \int \mathrm{f}(x) \, \mathrm{d}x \]
is the general solution for the above differential equation.

Note that because the indefinite integration of \(\mathrm{f}\) yields one arbitrary constant, it is congruous to the fact that this is a first order differential equation. In other words, every first order differential equation that has a solution will have infinitely many solutions owing to the one degree of freedom spanned by the existence of one arbitrary constant. Different values of this constant yield different solutions for the equation, hence deriving a family of solutions.

A particular solution of a differential equation is the one which satisfies particular instances of \(x\) and \(y\). In other words, substituting a given pairs of values of \(x\) and \(y\) into the general solution one can obtain the unknown constant \(C\) that arises from the indefinite integration.