Question

The differential equation of the family of curves $\mathrm{y}=\mathrm{\alpha cos}\left(\mathrm{x}+\mathrm{b}\right)$ is

• A. $\left(\frac{d^{2}y}{d{x}^2}\right)-y=0$
• B. $\left(\frac{d^{2}y}{d{x}^2}\right)+y=0$
• C. $\left(\frac{d^{2}y}{d{x}^2}\right)+2y=0$
• D. None of these

Answer 1

The correct option is B

$\left(\frac{d^{2}y}{d{x}^2}\right)+y=0$

Explanation for the correct option:

Given,

$\mathrm{y}=\mathrm{\alpha cos}\left(\mathrm{x}+\mathrm{b}\right)$

On differentiating w.r.t $x$

$\frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{\alpha sin}\left(\mathrm{x}+\mathrm{b}\right)$

Again,

$$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=-\mathrm{\alpha cos}\left(\mathrm{x}+\mathrm{b}\right) \\ \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=-\mathrm{y} \\ \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\mathrm{y}=0 \end{gathered}$$

Hence, the correct answer is option (B).