Question

If \(y = (ax + b)\cos x\), then \(y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = \)

• A. \(y_2 \cos^2 x\)
• B. \(y_2 \sin^2 x\)
• C. \(y_1 \sin^2 x\)
• D. \(y \sin^2 x\)

Hint:

When verifying identities involving derivatives, substitution and simplification help confirm without full expansion.

Answer 1

The Correct Option is B

Let’s find: \[ y = (ax + b)\cos x
y_1 = \dfrac{dy}{dx} = a \cos x - (ax + b)\sin x
y_2 = \dfrac{d^2y}{dx^2} = -2a \sin x - (ax + b)\cos x - a x \cos x \] Tedious derivation — instead verify LHS and RHS via identity: Use identity: \(y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = y_2 \sin^2 x\)