# Differential equations, separating variables!

• Feb 13th 2010, 10:16 AM
Yehia
Differential equations, separating variables!
if we have $\displaystyle dy/dx=e^(2x-1)$ how do we find the general solution??? help much appreciated!! thankyou!!
• Feb 13th 2010, 10:43 AM
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 $\displaystyle C\in \mathbb R$
• Feb 13th 2010, 10:58 AM
Yehia
Quote:

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!!!!!(Rofl)
• Feb 13th 2010, 11:04 AM
icemanfan
Quote:

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

$\displaystyle \cos x - \sin x = 0$

$\displaystyle \cos x = \sin x$

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

$\displaystyle \tan x = 1$
• Feb 13th 2010, 11:09 AM
Nyrox
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 $\displaystyle b$:

$\displaystyle V=\int_a^b\pi (f(x))^2 \, dx$
• Feb 13th 2010, 11:12 AM
ANDS!
Quote:

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!!!!!(Rofl)

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?
• Feb 13th 2010, 11:17 AM
Yehia
Quote:

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:

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

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