Formation Of Differential Equations

Forming a differential equation from a family of curves means eliminating the arbitrary constants by differentiating as many times as there are constants. Since y = A\cos x + B\sin x contains two independent constants A and B, we must differentiate twice to remove both. First, \frac{dy}{dx} = -A\sin x + B\cos x, which still carries the constants. Differentiating again gives \frac{d^2y}{dx^2} = -A\cos x - B\sin x. The crucial observation is that the right side is exactly the negative of the original y, since -A\cos x - B\sin x = -(A\cos x + B\sin x) = -y. Therefore \frac{d^2y}{dx^2} = -y, which rearranges to \frac{d^2y}{dx^2} + y = 0. Option \frac{d^2y}{dx^2} - y = 0 corresponds to the family A e^x + B e^{-x}, not trigonometric. Option \frac{dy}{dx} + y = 0 eliminates only one constant and is first order. Option \frac{d^2y}{dx^2} + \frac{dy}{dx} = 0 does not match the second derivative relationship. This is the standard JEE Advanced elimination technique. As a final consistency check, the resulting equation is second order with two constants eliminated, matching the requirement that order equals the number of arbitrary constants in the family.