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We know for $f(x)=0$ and $f(x)=e^x$ the function is same as its derivative. I was wondering if there are any other such functions and how do we go about finding it?

QFi
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Lipika Agrawal
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    The solution is unique, if you specify the value at a point, say $x=0$. Then the solution is $f(x)=f(0)e^x$. – lulu Mar 25 '17 at 23:40
  • Hint: To answer to your question, you need to solve: $\dfrac{d,f(x)}{dx}-f(x)=0$ – QFi Mar 25 '17 at 23:56

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If $f(x) \neq 0$, then you can look at $$ (\ln f)' = \frac{f'}{f} = 1\,, $$ and therefore $$ \ln f(x) = x + C_0\,,$$ for some constant $C_0$. Thus the function is of the form $$ f(x) = C e^x\,,$$ where $C = e^{C_0}$.

Martins Bruveris
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