1. ## differential equations

solve problem: dy/dx = e^(x+y)... (make final answer e.g. y= something)

After integrating 1/e^y with respect to dy and e^x with respect to dx, and without including the constant yet, i got
-e^(-y) = e^x
where -e^(-y) is from integration of 1/e^y with respect to dy.

If i write -e^(-y) + C = e^x, I will get final answer as y = -ln (k - e^x), where k = -C, which is the answer from the textbook.

Also if I write -e^(-y) = e^x + C, I will get the same correct answer.

1. My question is that.. is the C (constant) the combination of the constants of the integrals of LHS and RHS? And therefore we only add C to either side of the equation and not both?

2. No matter which side I place the C at for any differential equation problem, will I get the same result?

And lastly.. must I include a modulus sign after a log or ln? e.g. ln | 2+x | ?
or just ln (2+x) will do? Although 2+x must be a positive value..?

Please clear my doubts.. MILLION THANKS!

2. Hello, custer!

1. Is the $\displaystyle C$ the combination of the constants of the integrals of LHS and RHS? .Yes!
And therefore we only add $\displaystyle C$ to either side of the equation and not both? .Right!

2. No matter which side I place the $\displaystyle C$ for any differential equation problem,
will I get the same result? . Yes!
3. And lastly.. must I include a modulus sign after a log or ln?
e.g. $\displaystyle \ln| 2+x |$ . . . or just $\displaystyle \ln(2+x)$ will do?
If it is possible for the argument to be negative, absolute values should be used.

For your example, we should write: .$\displaystyle \ln|2+x|$

. . but for $\displaystyle \ln|x^2+4|$ the absolute values are redundant (but not incorrect).

3. Originally Posted by Soroban
Hello, custer!

If it is possible for the argument to be negative, absolute values should be used.

For your example, we should write: .$\displaystyle \ln|2+x|$

. . but for $\displaystyle \ln|x^2+4|$ the absolute values are redundant (but not incorrect).
I'd suggest that absolute value is only used when the logarithm arises from solving an integral. If the logarithm is a result of re-arranging an equation, absolute value is not used.