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Question
Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where a and b are arbitrary constants.
Solution
Given : `y = ae^(bx + 5)`
Differentiating y with respect to x.
`(dy)/(dx) = ae^(bx + 5) (b) = be^(bx + 5) = by` (Since `y= ae^(bx + 5)`) .....1
Differentiating (1) again with respect to x we get
`(d^2y)/(dx^2) = b (dy)/(dx)` .....(2)
Dividing (2) by (1) we get
`((d^2y)/(dx^2))/(dy/dx) = (b(dy/dx))/(by)`
`=> y (d^2y)/(dx^2) = ((dy)/(dx))^2`
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