Q.Verify that $y = A\sin x + B\cos x$, where $A, B$ are arbitrary constants, is the general solution of $\dfrac{d^2y}{dx^2} + y = 0$.

A differential equation typically has infinitely many solutions, all captured at once by its general solution — an expression containing exactly as many independent arbitrary constants as the order of the equation (a first-order equation has one arbitrary constant in its general solution, a second-order equation has two, and so on). For instance y = a sin x + b cos x, carrying two constants, is the general solution of a second-order equation, while y = a sin x alone or y = b cos x alone are each only particular solutions obtained by fixing one of the constants. A particular solution is any solution obtained from the general solution by assigning specific numerical values to its arbitrary constant(s), usually because extra information is given — an initial condition such as 'y = 2 when x = 0', or a point the solution curve must pass through. To VERIFY that a given expression is a solution of a stated differential equation, the expression is different …