What is the general equation of dy/dx =e^(x+y)?

2 Answers
Apr 5, 2018

y = ln(1/(C_0-e^x))

Explanation:

This is a separable differential equation so

dy/e^y = e^x dx or

-e^-y= e^x - C_0 or

y = ln(1/(C_0-e^x))

Apr 5, 2018

y = -ln|C-e^x|

Explanation:

We have:

dy/dx = e^(x+y)

Which we can write as

dy/dx = e^(x) e^(y)

We can collect terms for similar variables:

e^(-y) dy/dx = e^x

Which is a separable First Order Ordinary non-linear Differential Equation, so we can "separate the variables" to get:

int \ e^(-y) \ dy = int e^x \ dx

Both integrals are those of standard functions, so we can use that knowledge to directly integrate:

-e^(-y) = e^x - C

And we can readily rearrange for y:

e^(-y) = C-e^x

:. -y = ln|C-e^x|

Leading to the General Solution:

y = -ln|C-e^x|