Question

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)

Answer 1

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`