If $f'(x)=g(x)$ and $g'(x)= -f(x)$ for all $x$ and $f(2)=4=f'(2)$. Find value of $f^2(4) +g^2(4)$

Question

Problem:If $f'(x)=g(x)$ and $g'(x)= -f(x)$ for all $x$ and $f(2)=4=f'(2)$. Find value of $f^2(4) +g^2(4)$

Solution:

I tried this question by first principle method of derivative but didn't get the answer

Hint: Try to build $f''(x)$. What can you conclude? – Bernd Jan 14 '14 at 13:44 — Bernd, Jan 14 '14 at 13:44
Can you show $f^2(x)+g^2(x)$ is constant? – Empy2 Jan 14 '14 at 13:51 — Empy2, Jan 14 '14 at 13:51

score 1 · Accepted Answer · answered Jan 14 '14 at 13:59

1

HINT : use the fact that $\frac{d}{dx}(f^2) = 2f'(x)f(x)$

Using those you can prove that the expression $f^2(x) + g^2(x)$ is constant, this is conservation of energy in the physical system.

answered Jan 14 '14 at 13:59

user88595

then $f^2(4) + g^2(4) = f^2(2) + g^2(2)$ – rst Jan 14 '14 at 14:15
Yes, it's a constant. The idea is to this values does not change with respect to the variable. But you need to show it's a constant using my hint first. – user88595 Jan 14 '14 at 14:19
Then ans is $32$ .it is constant as derivative of $f^2(x) + g^2(x)$ is $0$. Am I right ? – rst Jan 14 '14 at 14:27
Exact. That's the quickest I see to solve it. You find the solution for $f$ and $g$ which is $\cos$ and $\sin$, reason why $f^2 + g^2 = cst$. But that takes longer... – user88595 Jan 14 '14 at 14:43

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