Since the given equation is separable, we can rewrite is as
![{\displaystyle {\frac {dy}{e^{y}}}=e^{3x}dx.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b57d5ca664670b00c8be0089e78fc1233ba0ce0c)
Taking integral on the both side of the equation, we have
![{\displaystyle \int e^{-y}dy=\int e^{3x}dx.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/56d1c27ee5e05f08d521a73facc09c0194020b45)
Note that using substitution
for any fixed number
, we have
![{\displaystyle \int e^{ax}dx=\int e^{u}{\frac {du}{a}}={\frac {1}{a}}e^{u}+C={\frac {1}{a}}e^{ax}+C.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/88641789434c146f605a122f7b5e915bf9e5630c)
Applying this for
and
, we obtain
![{\displaystyle \int e^{-y}dy={\frac {1}{-1}}e^{-y}+C_{1}=-e^{-y}+C_{1},}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/35e2e50dd80df330048ae27061d4b66129ff8fe7)
and
![{\displaystyle \int e^{3x}dx={\frac {1}{3}}e^{3x}+C_{2},}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/bacb9ba544449e519d75cbd06fd978b059d40618)
where
![{\displaystyle C_{1}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/babf569931f1a7b5182b9bec51873c2f5692fbb8)
and
![{\displaystyle C_{2}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/7ec545f7870665e1028b7492746848d149878808)
are arbitrary constants.
Therefore, we get
![{\displaystyle -e^{-y}+C_{1}=\int e^{-y}dy=\int e^{3x}dx={\frac {1}{3}}e^{3x}+C_{2},}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/28ad6da71ccf2eee67a2b30045a253aac6e20499)
which can be simplified as follows
![{\displaystyle e^{-y}=-{\frac {1}{3}}e^{3x}+C.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/fd866f041127557be528692c3cbb0f64698c55a5)
We plug
to find the constant
,
![{\displaystyle {\frac {1}{5}}=e^{-y(0)}=-{\frac {1}{3}}+C\iff C={\frac {8}{15}}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/86179b5078bcf3aaa38e8cd891f08b4d1cc3123b)
Finally, taking a logarithm to the simplified equation, we can find the solution
![{\displaystyle y=-\ln \left(-{\frac {1}{3}}e^{3x}+{\frac {8}{15}}\right).}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/1cefbf8f0d56253714b0fb1312795862e2d88848)
Answer: