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Question
Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.
Solution
Given: y = e2x (a + bx)
Differentiating the above equation, we get
`(dy)/(dx) = be^(2x) + 2 (a + bx)e^(2x)`
`= (dy)/(dx) = be^(2x) + 2y ...("i") [∵ y = e^(2x) (a + bx)]`
differentiating the above equation, we get
`(d^2y)/(dx^2) = 2 be^(2x) + 2(dy)/(dx)`
= `(d^2y)/(dx^2) = 2 ((dy)/(dx) - 2y) + 2(dy)/(dx) ...[∵ "from" ("i") "we get", be^(2x) = (dy)/(dx) - 2y]`
= `(d^2y)/(dx^2) = 4(dy)/(dx)- 4y`
Hence, the required differential equation is `(d^2y)/(dx^2) - 4 (dy)/(dx) + 4y= 0`.
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