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Question
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2) = 49y`
Solution
y = 500e7x + 600e-7x
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)`
= 500 e7x . 7 + 600 e-7x. (-7)
= 3500 e7x - 4200 e-7x
Differentiating again with respect to x,
`(d^2 y)/dx^2 = 3500 d/dx e^(7x) - 4200 d/dx e^(- 7x)`
`= 3500 xx e^( 7x) * 7 - 4200 e^(- 7x) (- 7)`
= 500 × 49 e7x + 600 × 49 e-7x
= 49(500 e7x + 600 e-7x)
= 49 y
∴ `(d^2y)/dx^2 = 49y.`
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