Question

Form the differential equation representing the family of curves y = e²x (a + bx), where 'a' and 'b' are arbitrary constants.

Answer 1

Given: y = e^2x (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`.