Q. The solution of the differential equation `dy/dx` = `e^(y + x) + e^(y - x)` is A) `e^-y = e^-x - e^x + c` B)` ` ```e^-y = e^x - e^-x + c` C)`e^-y = e^-x + e^x + c` D) None of these

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to separate the variables, such that:

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

Factoring out e^y yields:

`(dy)/(dx) = e^y*(e^x + 1/e^x)`

You need to divide by `e^y` both sides, such that:

`(dy)/(e^y(dx)) = (e^x + 1/e^x)`

You need to multiply by dx both sides, such that:

`(dy)/(e^y) = (e^x + 1/e^x)dx`

Integrating both sides, yields:

`int e^(-y)dy = int (e^x + 1/e^x)dx`

Using the property of linearity of integral yields:

`int e^(-y)dy = int (e^x)dx + int(1/e^x)dx`

`-e^(-y) = e^x - e^(-x) + c`

Hence, evaluating the solution to the given differential equation, yields `-e^(-y) = e^x - e^(-x) + c` , hence, you need to select the option D) none of these.