Suppose $f$ is defined on all real numbers. $f = f''$ and $f(0)=f'(0)=0$. Then show that $f=0$ for all $x$.
The following is what I did: Since we have $f=f''$. Then, multiply $f'$ on both sides: $f\cdot f'=f' \cdot f''$ $f \cdot f'-f' \cdot f''= \frac {1}{2}(f^2)'- \frac {1}{2}(f'^2)'$ This says, $f^2-f'^2=C$, and plug in number $x=0$, we can then say $C=0$. So, $f^2-f'^2=0$.
But, what is the next step? How can I show $f=0$ for all $x$?