A solution or an integral of a differential equation is a relation between the variables, not involving the differential coefficients, such that this relation and the derivatives obtained from it satisfy the given differential equation. For example\[y = A \cos x + B \sin x\] where \(A\) and \(B\) are any constants, is a solution of the differential equation \[\frac{{{d^2}y}}{{d{x^2}}} + y = 0\] as the differentiation of the given expression twice leads to \[\frac{{dy}}{{dx}} =  - A\sin x + B\cos x\,\]

\[\frac{{{d^2}y}}{{d{x^2}}} =  - A\sin x - B\cos x =  - y\]\[ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} + y = 0\] which is the given equation.