Question

A differential equation representing the family of curves $y=a\sin (\lambda x+\alpha )$ is:

• A. $\frac{d^{2}y}{d{x}^2}+{\lambda }^{2}y=0$
• B. $\frac{d^{2}y}{d{x}^2}-{\lambda }^{2}y=0$
• C. $\frac{d^{2}y}{d{x}^2}+{\lambda }^{}y=0$
• D. None of the above

Answer 1

The correct option is A $\frac{d^{2}y}{d{x}^2}+{\lambda }^{2}y=0$

$y=a\sin (\lambda x+\alpha )$

Differentiating the equation, we get

$\Rightarrow \frac{dy}{dx}=a\lambda \cos (\lambda x+\alpha )$

Differentiating the equation

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-a{\lambda }^2\sin (\lambda x+\alpha )=-{\lambda }^{2}y$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+{\lambda }^{2}y=0$
Hence, option A is correct.