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Possible Duplicate:
Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$

Here's a question I got for homework:

Let f a differentiable function such that $f(x)=f'(x)$ for all $x$. Prove that there exist a $c \in \mathbb{R}$ such that $f(x) = c \cdot e^x$

Hint: notice $\dfrac{f(x)}{e^x}$

So, as it turns out this hint was not enough.

Any more hints? Thanks!

Community
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yotamoo
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2 Answers2

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HINT:

What's the derivative of $\dfrac{f(x)}{e^x}$ (With the quotient rule, perhaps it's easier to see). Then what does that mean?

davidlowryduda
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It's a first-order linear ordinary differential equation. To put it in a simple way, let $y=f(x)$. Then $$ \frac{dy}{dx}=y. $$ Hence,
$$ \frac{dy}{y}=dx, $$ (if $y$ doesn't equal $0$).

Integrating in both sides, we get $$ \ln|y|=x+c_1, $$ where $c_1$ is a constant. Therefore， $$ |y|=e^{x+c_1}=e^{c_1}*e^x. $$ Let $|c|=e^{c_1}$, then we get $$ y=c*e^x. $$ If $y=0$, then it of course satisfies the condition.

Pan Yan
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    Sorry for the bad notation. Just new here, and I am still learning how to use the mathematical notation in this website. – Pan Yan May 22 '12 at 08:42
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    I found a useful way to see how people generate symbols here is to click edit and see how they write thing (while not editing the answer of course!) – Holdsworth88 May 22 '12 at 08:44
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    @Holdsworth88: If you right-click on a formula, select Show Math As, and then select TeX Commands, you’ll get a pop-up window showing the $\LaTeX$ that was actually used. – Brian M. Scott May 22 '12 at 09:00
  • @BrianM.Scott Well, that simplifies that process a bit. Thank you for telling me about that feature. – Holdsworth88 May 22 '12 at 09:04
  • @[email protected] Thank you for telling me that. – Pan Yan May 22 '12 at 09:06
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    What if $y$ is not the zero function but happens to take the value zero at some points? Also, the constant $c$ needn't be positive, though the argument seems to imply that it is. – Shane O Rourke May 22 '12 at 10:48
  • @ Shane O Rourke The derivative of ln|y| is 1/y, so when you look back to the answer again which I have edited, you will find c can be negative. And it can't be the case that y happens to take the value zero at some points but not the zero function.(here y is continuous at every x) – Pan Yan May 22 '12 at 11:55