Question

If y = (A + Bx)e^−2x, prove that `(d^2y)/(dx^2) + 4 dy/dx + 4y = 0`.

Answer 1

Given, y = (A + Bx)e^−2x

Differentiating w.r.t x:

⇒ `dy/dx = (A + Bx)(-2e^(-2x)) + e^(-2x)B`

⇒ `dy/dx = -2(A + Bx)e^(-2x) + Be^(-2x)`

⇒ `dy/dx = (-2y) + Be^(-2x)`

⇒ `dy/dx + 2y = Be^(-2x)`   ...(i)

Differentiate (i) again w.r.t x:

`d/dx(dy/dx + 2y) = d/dx(Be^(-2x))`

⇒ `(d^2y)/(dx^2) + 2 dy/dx = -2Be^(-2x)`   ...(ii)

Use (i) in (ii):

From (i), `dy/dx + 2y = Be^(-2x)`

⇒ `(d^2y)/(dx^2) + 2 dy/dx = -2 (dy/dx + 2y)`

⇒ `(d^2y)/(dx^2) + 2 dy/dx + 2 dy/dx + 4y = 0`

⇒ `(d^2y)/(dx^2) + 4 dy/dx + 4y = 0`

Hence proved.