A Taylor series

i was wondering if you could help me out with this solution.... -- Given the following information of the function

f''(x) = 2f(x) for every value of x
f(0) = 1
f(0) = 0

what is the complete Taylor series for f(x) at a = 0

Hi Karan,

I think that you want condition 3. to be f'(0) = 0

If you are to write the Taylor series that you have to assume that f(x) can de differentiated as many times as you like. Condition 1. then contains a great deal of information. First

f''(0) = 2f(0) = 2
Now if you differentiate both sides of condition 1. you get

f'''(x) = 2 f'(x)
and hence

f'''(0) = 2 f'(0) = 2 (0) = 0
Differentiating again

f⁽⁴⁾(x) = 2 f''(x)
and hence

f⁽⁴⁾(0) = 2 f''(0) = 2 [2f(0)] = 2²

Following this procedure you can find f⁽ⁿ⁾(0) for all n and hence you can construct the Taylor series.

Cheers,
Harley Go to Math Central