General and Particular Solutions of a Differential Equation

A function $y = \phi(x)$ is called a solution of a differential equation if, when $y$ and its derivatives (computed from $\phi$) are substituted into the equation, the equation is satisfied identically -- i.e. it becomes a true statement for every $x$ in the domain considered, not merely for special values of $x$.

Verifying a solution. To check that a given $y = \phi(x)$ solves a differential equation of order $n$: differentiate $\phi$ exactly $n$ times, substitute $y, y', y'', \ldots, y^{(n)}$ into the left-hand side of the equation, and confirm the result equals the right-hand side identically (see Examples 3–4).

General solution. A differential equation of order $n$ typically has a family of solutions containing $n$ independent arbitrary constants; such a family is called the general solution (sometimes "primitive"). For instance, $y = A\sin x + B\cos x$ (two arbitrary constants $A, B$) is the general solution of the order-$2$ equation $\dfrac{d^2y}{dx^2} + y = 0$ -- every choice of $A, B$ gives a solution, and every solution arises from some choice of $A, B$.

Particular solution. Assigning specific numerical values to the arbitrary constants of a general solution -- most often by requiring the solution to pass through a given point, or to satisfy a stated condition -- singles out one member of the family, called a particular solution. …