A separable equation

Calculus Level 1

If $\dfrac{dy}{dx}=e^{x+y}$ , and $y(0)=0$ , then the value of $y(-\ln 2)$ can be written in the form $\ln \left(\dfrac mn\right)$ , where $m$ and $n$ are coprime positive integers . Find $m+n$ .

The answer is 5.

1 solution

Samir Khan
Jun 14, 2016

Separating the variables gives $e^{-y}\, dy= e^x\, dx\implies \int e^{-y}\, dy=\int e^x\, dx\implies -e^{-y}= e^x+C.$ Plugging in the initial condition, we find $C=-2$ , so $-e^{-y}=e^x-2$ or $y=-\ln(2-e^x)$ . Then, $y(-\ln 2)=\ln(3/2)$ , so the answer is 5.

Then why is the correct answer being shown as $8$ ?

Rishabh Jain - 5 years ago

That was an error on my part; it should be fixed now. Sorry about that.

Samir Khan - 5 years ago

How could you change the answer? Are you a staff member( just asking)? Bcoz I'm pretty sure that I entered 5, was shown incorrect and then revealed the solutions to see 8 is the answer but now it shows 5!!

Rishabh Jain - 5 years ago

@Rishabh Jain – Thanks. I've given credit for those who has previously answered 5.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “line line line” menu in the top right corner. This will notify the problem creator who can fix the issues.

Brilliant Mathematics Staff - 5 years ago

@Brilliant Mathematics – Obviously.... I had already filed a report..

Rishabh Jain - 5 years ago

I believe you mean that $y(-\ln(2) ) = y (\ln(1/2))=-\ln(2-1/2)=-\ln(3/2) = \ln(2/3)$ no?

Chris Callahan - 4 years, 11 months ago

A separable equation

The answer is 5.

1 solution

0 pending reports