# Thread: Differential equations, separating variables!

1. ## Differential equations, separating variables!

if we have $\displaystyle dy/dx=e^(2x-1)$ how do we find the general solution??? help much appreciated!! thankyou!!

2. It's a matter of simple integration. You know that $\displaystyle [e^x]'=e^x$, so

$\displaystyle [\frac{1}{2}e^{2x-1}]'=e^{2x-1}$

A general solution is thus

$\displaystyle y(x)=\frac{1}{2}e^{2x-1}+C$ (with $\displaystyle C\in \mathbb R$

3. Originally Posted by Nyrox
It's a matter of simple integration. You know that $\displaystyle [e^x]'=e^x$, so

$\displaystyle [\frac{1}{2}e^{2x-1}]'=e^{2x-1}$

A general solution is thus

$\displaystyle y(x)=\frac{1}{2}e^{2x-1}+C$ (with $\displaystyle C\in \mathbb R$
Yes thank you!! that's right, can you just help me with 2 other problems?

firstly, if i have the function cosx-sinx=0 how do i solve it? im stumped :/

and secondly how to i find the volum of space if the function x^(1/2)*e^x revolved around the x axis 360 degrees with the limits 1 and 2?

Thanks a million!!!!!

4. Originally Posted by Yehia
firstly, if i have the function cosx-sinx=0 how do i solve it? im stumped :/
$\displaystyle \cos x - \sin x = 0$

$\displaystyle \cos x = \sin x$

$\displaystyle \frac{\sin x}{\cos x} = 1$

$\displaystyle \tan x = 1$

5. These have nothing to do with your original question... But anyway, for the first one, plot the graphs of both sine and consine, and you'll get your answer. As for the second, you must've seen the connection between the integral and a solid obtain by revolution. Calling $\displaystyle V$ the volume of the solid obtained by revolving a curve $\displaystyle f(x)$ around the $\displaystyle x$ axis between points $\displaystyle a$ and $\displaystyle b$:

$\displaystyle V=\int_a^b\pi (f(x))^2 \, dx$

6. Originally Posted by Yehia
Yes thank you!! that's right, can you just help me with 2 other problems?

firstly, if i have the function cosx-sinx=0 how do i solve it? im stumped :/

and secondly how to i find the volum of space if the function x^(1/2)*e^x revolved around the x axis 360 degrees with the limits 1 and 2?

Thanks a million!!!!!
Already answered on the first one; on the volume question, I would use integration by parts - but it looks nasty. Is this equation written correctly?

7. Originally Posted by ANDS!
Already answered on the first one; on the volume question, I would use integration by parts - but it looks nasty. Is this equation written correctly?
yes it is but im unfamiliar with writing it on these threads :S ill try:

$\displaystyle f(x)=x^{1/2}e^x$

$\displaystyle f(x)=x^{1/2}e^x$