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प्रश्न
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
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उत्तर
y = 500e7x + 600e–7x ...(1)
On differentiating with respect to x,
`dy/dx = d/dx (500 e^(7x) + 600 e^(- 7x))`
= `500 d/dx e^(7x) + 600 d/dx e^(- 7x)`
= `500 e^(7x) d/dx (7x) + 600 e^(- 7x) d/dx (-7x)`
= 500e7x . 7 + 600e–7x. (−7)
= 3500e7x − 4200e−7x
Differentiating again with respect to x,
`(d^2 y)/dx^2 = d/dx [3500e^(7x) − 4200e^(−7x)]`
= `3500 d/dx e^(7x) - 4200 d/dx e^(- 7x)`
= `3500e^(7x) d/dx (7x) - 4200e^(- 7x) d/dx (- 7x)`
= 3500e7x . 7 − 4200e−7x . (−7)
= 24500e7x − 29400e−7x
= 500 × 49e7x + 600 × 49e−7x
= 49(500e7x + 600e−7x)
= 49 y ...[From equation (1)]
∴ `(d^2y)/dx^2` = 49y
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