Question

Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where a and b are arbitrary constants.

Answer 1

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`