# Thread: differential equations

1. ## differential equations

Hi all,

I need help solving for this:
$\displaystyle \int \frac{du}{3u + \sqrt{u}} = \int \frac{dx}{x}$

Thanks in advance
ArTiCk

2. Originally Posted by ArTiCK
Hi all,

I need help solving for this:
$\displaystyle \int \frac{du}{3u + \sqrt{u}} = \int \frac{dx}{x}$

Thanks in advance
ArTiCk
Let $\displaystyle \xi=\sqrt{u}\Rightarrow{\xi^2=u}$

So $\displaystyle du=2\xi{d\xi}$

So we have

$\displaystyle \int\frac{2\xi}{3\xi^2+\xi}d\xi=\frac{2\ln(3\xi+1) }{3}$

back subbing we get

$\displaystyle \frac{2\ln(3\sqrt{u}+1)}{3}=\ln((3\sqrt{u}+1)^{\fr ac{2}{2}})$

The right side is obviously $\displaystyle \ln(x)$

so we have

$\displaystyle \ln((3\sqrt{u}+1)^{\frac{2}{3}})=\ln(x)$

exponentiating both sides gives

$\displaystyle (3\sqrt{u}+1)^{\frac{2}{3}}=x$

and I assume this was solving for x

so done

3. Hello, ArTiCK!

$\displaystyle \int \frac{du}{3u + \sqrt{u}} \:= \:\int \frac{dx}{x}$
Let $\displaystyle \sqrt{u} \,=\,v\quad\Rightarrow\quad u \,=\,v^2\quad\Rightarrow\quad du \,=\,2v\,dv$

Substitute: .$\displaystyle \int\frac{2v\,dv}{3v^2+v} \;=\;\int\frac{dx}{x} \quad\Longrightarrow\quad 2\int\frac{dv}{3v+1} \:=\:\int\frac{dx}{x}$

Go for it!

4. Isomorphism also said few words here.