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)          .....(1)
Differentiating both sides, we get

\[\frac{dy}{dx} = a \cos\left( x + b \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - a \sin\left( x + b \right) \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - a \times \frac{y}{a} ...............\left[\text{Using (1)} \right]\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - y \]
\[ \Rightarrow \frac{d^2 y}{d x^2} + y = 0\]