# Thread: Integration / Splitting the variables

1. ## [SOLVED] Integration / Splitting the variables

Hey, I've gotta brush up on my integration / differentiation before the september term and have come across a question that's giving me trouble (well.. a few tbh but I might go back to them later..)

$\displaystyle \frac{\delta x}{\delta t} = 3x$

solve, given x=10, t=0
Cut-down workings:

rearrange and differentiate with respect to x
$\displaystyle \int (\frac{1}{3} \frac{\delta x}{\delta t}) \delta x = \int (3x) \delta x$

And this simplifies /rearranges to:
$\displaystyle x = e^{3t+c}$

Problem
According to the textbook, the answer should be:
$\displaystyle x = e^{3t+10}$

The closest I've gotten so far is as follows:

$\displaystyle \ln (x) = \ln e^{3t+c}$
$\displaystyle \ln (x) = 3t+c$

t is given in the question as 0, therefore:
$\displaystyle c = \ln (x)$

if $\displaystyle c = \ln (A)$, then
$\displaystyle \ln (A) = \ln (x)$

x is given as 10, therefore:

$\displaystyle A = 10$

However, I'm probably going at it slightly wrong since I can't get A=10 to become c=10 ...

Any help?

Thanks,
kwah

2. Originally Posted by kwah

Problem
According to the textbook, the answer should be:
$\displaystyle x = e^{3t+10}$
the text "solution" does not match up with the given initial condition ...
x = 10 when t = 0

I also get $\displaystyle x = 10e^{3t}$ ... maybe they meant to type $\displaystyle x = e^{3t + \ln{10}}$

3. thanks... i think its just an error in the textbook =]