Question

The differential equation for $y=A\cos \alpha x+B\sin \alpha x$, where $A$ and $B$ are arbitrary constants is

• A. $\frac{d^{2}y}{d{x}^2}-{\alpha }^{2}y=0$
• B. $\frac{d^{2}y}{d{x}^2}+{\alpha }^{2}y=0$
• C. $\frac{d^{2}y}{d{x}^2}+{\alpha }^{}y=0$
• D. $\frac{d^{2}y}{d{x}^2}-{\alpha }^{}y=0$

Answer 1

Answer: (B)

$\frac{d^{2}y}{d{x}^2}+{\alpha }^{2}y=0$

$y=A\cos \alpha x+B\sin \alpha x$ .....(i)

$\frac{dy}{dx}=-A\alpha \sin \alpha x+B\alpha \cos \alpha x$.....(ii)

$\frac{d^{2}y}{d{x}^2}=-A{\alpha }^2\cos \alpha x-B{\alpha }^2\sin \alpha x$ .....(iii)

$\therefore {\alpha }^2\times \left(i\right)+\left(ii\right)$

$\Rightarrow {\alpha }^{2}y+\frac{d^{2}y}{d{x}^2}=0$