If f''(x)= -f(x) and g(x)= f'(x) and F(x) = (f(x/2))^2 + (g(x/2))^2 and given that F(5) = 5 then F(10) equals ? (hint : answer is an integer from 0 - 9)

1 Answer
Ananda Dasgupta
May 14, 2018

5

Explanation:

Let G(x) equiv (f(x))^2+(g(x))^2. Then

{dG}/dx = 2f(x) f^'(x)+2g(x) g^'(x)
qquad = -2f''(x) g(x) + 2g(x) f''(x) = 0

where we have used f''(x) = -f(x), f^'(x) = g(x)and g^'(x) = f''(x)

Thus G(x) is a constant, and so is F(x)=G(x/2).

Since F(5) = 5, we must have F(10)=5

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