Analytically determine if $f(x) = f'(x)$ is possible?

Question

I was taking a test and two true/false type questions were asked.

In one of them, I had to say if there is a function $f(x)$ such that $f(x) = f'(x)$. Of course, $e^x$ is such a function and almost everyone who has taken a calculus course knows this fact well.

In the other question, I had to determine if $f(x) = -f'(x)$ was possible.

I was completely stumped at this one. I had never before encountered a function with such property nor did I know how to approach this problem analytically as I am just a high school student.

My question is: is there an analytical way to determine if such a function exists? By analytical, I mean no guessing allowed and just giving an example won't be enough.

Is this possible? If not, can you give an example of a function with the above property?

Answer 1

If $f'(x)=f(x)$ and if $g(x)=f(-x)$, then $g'(x)=-f'(-x)=-f(-x)=-g(x)$. Can you take it from here?

Answer 2

Linear differential equations $ay"+by'+cy$ are solved by considering the roots of the polynomial $ax^2+bx+c$, here you have $x+1=0$.

Answer 3

My question is: is there an analytical way to determine if such a function exists?

There's a theorem for that. Specifically, the existence-uniqueness theorem for differential equations.

Wikipedia link

Answer 4

As you're in high school, you have probably not covered the topic of differential equations but you can use one to find the analytical solution to your question. $dy/dx + y = 0$ and you will find that the solution is $y=ce^{-x}$ (where c is any constant): https://www.wolframalpha.com/input/?i=dy%2Fdx+%2B+y+%3D+0