Absolute Values in Separable Differential Equations

[patrickbotros]

When solving a separable differential equation, my textbook says this:
ln|v-49|=-t/5+C→
|v-49|=e^-t/5+C→
v=49+ce^-t/5
What happened to the absolute values? I think it has something to do with the exponential always being positive.

 

[SammyS]

When solving a separable differential equation, my textbook says this:
ln|v-49|=-t/5+C→
|v-49|=e^-t/5+C→
v=49+ce^-t/5
What happened to the absolute values? I think it has something to do with the exponential always being positive.

The equation $\displaystyle\ |v-49|=e^{-t/5+C}\$ is equivalent to

$\displaystyle\ v-49=\pm e^{-t/5+C}\$ → $\displaystyle\ v-49=c\, e^{-t/5}\$, where $\ c = \pm\ln(C)$​

So the new constant, c, (lower case) absorbs the ± .

 

[patrickbotros]

The equation $\displaystyle\ |v-49|=e^{-t/5+C}\$ is equivalent to

$\displaystyle\ v-49=\pm e^{-t/5+C}\$ → $\displaystyle\ v-49=c\, e^{-t/5}\$, where $\ c = \pm\ln(C)$​

So the new constant, c, (lower case) absorbs the ± .

Ohhhh. HAHA! That was dumb :)