Question

Use separation of variables to find the general solution of the differential equation.
e^x (y' + 7) = 1
So I'm stumped here, do I move the y'+7 to the other side, or do I distribute the e? I Know I need to separate the x and y, I'm just very confused..

Answer 1

in order to separate the y' from x; try taking ln on both left n right side of the eqn...

Answer 2

\(\large\color{black}{ \displaystyle e^x\left(\frac{dy}{dx}+7\right)=1 }\)

\(\large\color{black}{ \displaystyle \left(\frac{dy}{dx}+7\right)=\frac{1}{e^x} }\)

\(\large\color{black}{ \displaystyle \frac{dy}{dx}=\frac{1}{e^x}-7 }\)

\(\large\color{black}{ \displaystyle \color{red}{\int}\frac{dy}{dx}\color{red}{dx}=\color{red}{\int}\left(\frac{1}{e^x}-7\right)\color{red}{dx} }\)

Answer 3

no need to ln both sides, the left side is simply y+C
and the right side you can do.

Answer 4

wait - sorry I misread the eqn. i think @SolomonZelman has the right solution here.

Answer 5

@SolomonZelman you're a math wiz haha thank you so much, and just to make sure y' is the same as dy/dx right? And thank you @sdfgsdfgs!

Answer 6

well, dy/dx is a more convenient for me notation... also you can visualize how to cancel the differential.

Answer 7

the derivative is the slope after-all, that is: \(\large\color{black}{ \displaystyle \Delta y / \Delta x }\) - and that is \(\large\color{black}{ \displaystyle {\rm d} y / {\rm d} x }\)

Answer 8

well, the notation doesn't matter that much, but I like it that way.

Answer 9

I guess you are done pretty much, bye and yw

Answer 10

Got it, just making sure, thank you again! :-)

Answer 11

anytime