# I Is the exponential function, the only function where y'=y?

#### Phylosopher

Hello,

I was wondering. Is the exponential function, the only function where $y'=y$.

I know we can write an infinite amount of functions just by multiplying $e^{x}$ by a constant. This is not my point.

Lets say in general, is there another function other than $y(x)=ae^{x}$ ($a$ is a constant), where $\frac{dy}{dx}=y$.

I would really appreciate it if we can work a proof.

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#### andrewkirk

Homework Helper
Gold Member
Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}

#### Phylosopher

Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}
Exactly. But I was thinking. Is this enough to say that it is the only solution?

I know this sounds hilarious, but when I studied calculus no one pointed out that the solution is unique. Is it?

#### lurflurf

Homework Helper
we have
$\dfrac{d}{dx}y=y$
let us rewrite it as
$\dfrac{d}{dx}e^{-x}y=0$
now consider
$\dfrac{d}{dx}u=0$
what can we say about u?

#### Phylosopher

we have
$\dfrac{d}{dx}y=y$
let us rewrite it as
$\dfrac{d}{dx}e^{-x}y=0$
now consider
$\dfrac{d}{dx}u=0$
what can we say about u?
Constant w.r.t x

#### lurflurf

Homework Helper
^then the only solutions are
$e^{-x}y=C$
as desired

Homework Helper
Gold Member
2018 Award

#### Phylosopher

^then the only solutions are
$e^{-x}y=C$
as desired

I will give it a try and read it. Thanks

#### FactChecker

Gold Member
2018 Award
Exactly. But I was thinking. Is this enough to say that it is the only solution?
Step-by-step, there are no alternatives except for the choice of C and D.

#### fresh_42

Mentor
2018 Award
Is this enough to say that it is the only solution?
You can use Picard-Lindelöf.

#### Math_QED

Homework Helper
Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}
You divided by $y$, which is not allowed when $y = 0$.

This happens to be a solution, which can be included if you take $y = A\exp(x)$ with $A \geq 0$ as general solution (instead of $A > 0)$

#### mathwonk

Homework Helper
to expand on lurflurf's solution, use the quotient rule to show that if y'=y, then (y/e^x)' = 0, hence y/e^x = constant.

using this a lemma, one can deduce that the kernel of the linear constant coefficient operator P(D), where Df = f' and P is a polynomial of degree n, has dimension n. see my web notes on linear algebra, last section of chapter 5, roughly page 119:

http://alpha.math.uga.edu/~roy/laprimexp.pdf

briefly, the argument above gives the kernel of the operator (D-1) which generalizes to that of (D-c) and then by factoring a polynomial P(D) into products of linear factors of form (D-c), we get the result.

Last edited:

"Is the exponential function, the only function where y'=y?"

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