The differential equation of the family of curves \(y=a\sin(bx+c)\), a and c are parameters, is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{d^2y}{dx^2}+b^2y=0\)
The family has two arbitrary constants, a and c (b is a fixed number), so its differential equation is of order 2. It comes from differentiating twice and eliminating a and c.
\(y = a\sin(bx+c)\)
\( \displaystyle \frac{dy}{dx} = ab\cos(bx+c) \)
\( \displaystyle \frac{d^2y}{dx^2} = -ab^2\sin(bx+c) \)
Since \(a\sin(bx+c) = y\):
\( \displaystyle \frac{d^2y}{dx^2} = -b^2y \)
\( \displaystyle \frac{d^2y}{dx^2}+b^2y = 0 \)