Q.The differential equation representing the family of curves $y=\mathrm{A}\cos(x+\mathrm{B})$, where $\mathrm{A}$ and $\mathrm{B}$ are parameters, is

A differential equation can be manufactured from any family of curves (or functions) that carries arbitrary constants, by eliminating those constants — and, conversely, verifying that a given expression is a solution of a stated differential equation is the reverse check of the same idea.

The elimination method. Suppose a family of curves is written with $n$ arbitrary constants. To form the differential equation that this whole family satisfies (and that no longer contains any of those constants):

1. Differentiate the defining equation successively $n$ times, producing $n$ new equations.
2. Together with the original equation, that gives $(n+1)$ equations.
3. Eliminate the $n$ arbitrary constants from these $(n+1)$ equations algebraically.
4. The result is a differential equation of order $n$ — order exactly matches the number of constants eliminated: one constant gives a first-order equation, two constants give a second-order equation, and so on.

This is genuinely an elimination problem, not a differentiation recipe alone — after differentiating, the constants are isolated and substituted back (or the several equations are combined) until every trace of $\mathrm A$, $\mathrm B$, $a$, $b$, … is gone and only $x,y$ and derivatives of $y$ remain.

Straight from a physical law. Many differential equations are not formed this way at all — they are simply the direct mathematical translation of a stated rate relationship, with no constants to eliminate. "The rate of change of $Q$ is proportional to $Q$" becomes $\dfrac{dQ}{dt}=kQ$ immediately; "proportional to $A$ and inversely proportional to $B^2$" becomes $\dfrac{dA}{dB}=\dfrac{kA}{B^2}$; and so on. Newton's second law for a falling body, $m\dfrac{d^2h}{dt^2}=f\!\left(t,h,\dfrac{dh}{dt}\right)$, and the population models $\dfrac{dN}{dt}=rN$ (Malthusian growth) and $\dfrac{dN}{dL}=kN(L-N)$ (logistic growth) are built this way, straight out of the stated law, with no family of curves or constants involved at all.

Verifying a solution. Given a candidate expression $y=\phi(x)$ (possibly with arbitrary constants) and a target differential equation, substitute $y$ and its derivatives (found by differentiating $\phi$ the required number of times) into the equation and confirm the two sides become identical. This is exactly the reverse direction of elimination: if $\phi$ has $n$ arbitrary constants and satisfies an order-$n$ equation, it is that equation's general solution.