Explain how to get the right solution of y $dy/dx=y$

Question

When solving the following equation to find y as a function of x:

\begin{equation} dy/dx=y \end{equation}

First I divide both sides by $y$ and multiply both sides by $dx$:

$dy/y=dx$

Then I integrate both sides:

$\ln(y)+C1=x+C2$

Then:

$y=e^{x+C2-C1}=e^{x+C}$

I now know my result is problematic becomes $y=-e^x$ also satisfies $dy/dx=y$ but it is included in my solution: $y=e^{x+C}$. So can you help me find what is wrong with my analysis process and provide the right analysis process.

Thanks.

Update: I said $y=-e^x$ is not included in $y=e^{x+C}$ because $e^C>0$.

I do not understand your problem. $y = \exp(x+C)$ is the general solution. For $C = i,\pi$ this becomes $y = - \exp(x)$. — Martin R, Oct 04 '14 at 21:11
This procedure yields the ${\it form}$ of the answer. Constants are, as always, free to vary. — user795305, Oct 04 '14 at 21:13
@MartinR What I am confused is that normally $e^C>0$. I did not expect $C$ to be a complex number. Is there an alternative derivation process that assume all numbers are real and can still produce $y=Ce^x$ where C can be positive or negative? — Changwang Zhang, Oct 05 '14 at 07:40

score 1 · Answer 1 · answered Oct 05 '14 at 00:38

1

It should be noted that $$\int \frac{dy}{y} =\ln|y| + C$$ That absolute value is important.

This means that $|y| = De^x$ for some positive constant $D$.

The really odd trick is that $D=0$ works, as well...

answered Oct 05 '14 at 00:38

Thomas Andrews

177,126

score -2 · Answer 2 · answered Oct 04 '14 at 21:12

-2

$$\frac{dy}{dx} = y \iff$$ $$\frac{1}{y} dy = dx \iff$$ $$ \int \frac{1}{y} dy = \int dx \iff $$ $$ \ln(y) + C_1 = x+C_2 \iff$$ $$ \ln(y) = x+C \iff$$ $$e^{\ln(y)} = e^{x+C} \iff$$ $$e^{\ln(y)} = e^{x}e^{C} \iff$$

$$y = D e^x$$

where $D$ may be chosen to satisfy the initial conditions:

For example, if $y(0) = \pi$, then $y(0) = De^0 = \pi \implies D=\pi$

answered Oct 04 '14 at 21:12

Squirtle

6,698

2

Did you read the question? He is confused about why $y=-e^x$ is also a solution. Traditionally, $D=e^C>0$... – Thomas Andrews Oct 04 '14 at 21:24

Explain how to get the right solution of y $dy/dx=y$

2 Answers2