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प्रश्न
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
विकल्प
`("d"^2y)/("d"x^2) - alpha^2y` = 0
`("d"^2y)/("d"x^2) + alpha^2y` = 0
`("d"^2y)/("d"x^2) + alphay` = 0
`("d"^2y)/("d"x^2) - alphay` = 0
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उत्तर
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is `("d"^2y)/("d"x^2) + alpha^2y` = 0.
Explanation:
Given equation is : y = A cos a x + B sin a x
Differentiating both sides w.r.t. x, we have
`("d"y)/("d"x) = -"A" sin alpha x * alpha + "B" cos alpha x * alpha`
= `- "A" alpha sin alphax + "B" alpha cos alpha x`
Again differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = -"A"alpha^2 cos alpha x - "B" alpha^2 sin alpha x`
⇒ `("d"^2y)/("d"x^2) = -alpha^2 ("A" cos alphax + "B" sin alpha x)`
⇒ `("d"^2y)/("d"x^2) = - alpha^2y`
⇒ `("d"^2y)/("d"x^2) + alpha^2y` = 0
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