Differentiation question with continuity

[inter060708]

Homework Statement

Suppose a function f is continuous and has continuous derivatives of all orders for all
x. it satisfies xf ''(x) + f '(x) + xf(x) = 0. Given f(0) = 1

find the value of f '(0) and f '' (0).

Homework Equations

The Attempt at a Solution

when x=0,

0f''(0) + f ' (0) + 0f(0) = 0
therefore f ' (0) = 0

Now, I am not sure on how to find f ''(0). Is it undefined?
Thanks for your help.

 

[Dick]

There's no attempt at a solution. You have to make one. Then you can get some help.

 

[inter060708]

I have added the solution attempt, please help. Thanks :)

 

[Dick]

That helps. Try differentiating xf ''(x) + f '(x) + xf(x) = 0 and putting x=0 again.

 

[inter060708]

Wow. Thanks Dick! :D