Real analysis question about differentiation
Assume that $f$ and $f'$ are differentiable on $\mathbf R$ and that for every x in $\mathbf R$, $f(x) + f''(x) = 0 $. Show that $g (x)= f^2(x) + (f'(x))^2$ is a constant.
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Suppose that $f:\Bbb R\to\Bbb R$ and $f+f''=0$.
Then $g=f^2+(f')^2$ is constant.
It's amusing to note that this proves uniqueness for the solution:
Cor. If $f(0)=f'(0)=0$ then $f=0$.
Cor. If $f''+f=0$, $f(0)=a$ and $f'(0)=b$ then $f(x)=a\cos(x)+b\sin(x)$.
Proof: The previous corollary shows that $g=0$, if $g(x)=f(x)-(a\cos(x)+b\sin(x))$.