The differential equation of the family of curves y=asin(bx+c), a and…

The correct answer is option (1) → $\frac{d^2y}{dx^2}+b^2y=0 $

$y=a\sin(bx+c)$

so $\frac{dy}{dx}=ab\cos(bx+c)$

so $\frac{d^2y}{dx^2}=-b^2a\sin(bx+c)=-b^2y$

so $\frac{d^2y}{dx^2}+b^2y=0$