Question

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.

Answer 1

We have,
y = a sin (x + b) .....(2)
Differentiating both sides, we get
$$\begin{gathered} \frac{dy}{dx}=a\cos \left(x+b\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-a\sin \left(x+b\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-a\times \frac{y}{a}\left[\mathrm{Using}\left(2\right)\right] \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-y \\ \Rightarrow \frac{d^{2}y}{d{x}^2}+y=0 \end{gathered}$$