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प्रश्न
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
उत्तर
We have,
y = a sin (x + b) .....(1)
Differentiating both sides, we get
\[\frac{dy}{dx} = a \cos\left( x + b \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - a \sin\left( x + b \right) \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - a \times \frac{y}{a} ...............\left[\text{Using (1)} \right]\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - y \]
\[ \Rightarrow \frac{d^2 y}{d x^2} + y = 0\]
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