Question

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be ^–8x, where A and B are arbitrary constants

Answer 1

Given y = Ae^8x + Be^–8x  .........(1)

Where A and B are arbitrary constants differentiating equation (1) twice successively because we have two arbitrary constant, we get

`("d"y)/("d"x)` = Ae^8x + Be^–8x (– 8)

`("d"y)/("d"x)` = 8Ae^8x – 8Be^–8x  ........(2)

`("d"^2y)/("d"x^2)` = 8Ae^8x (8) – Be^–8x (– 8)

= 64Ae^8x + 64Be^–8x

= 64[Ae^8x + Be^–8x]  .........(3)

Substituting equation (1) in eqn (3), we get

`("d"^2y)/("d"x^2)` = 64y

`("d"^2y)/("d"x^2) - 64y` = 0 is the required differential equation.