variable substitution question(diff)

**transgalactic** · August 16th 2009, 01:06 PM

$\frac{xy'}{(\ln x\arctan y)-1}=(1+y^2)\arctan y\\$
$t=\arctan y\\$
$t'=\frac{1}{1+y^2}y'\\$
$\frac{x}{\ln (x)t-1}=\frac{t}{t'}\\$

$\frac{x}{\ln (x)t-1}=\frac{tdx}{dt}\\$
$xdt=(\ln (x)t-1)tdx\\$
$\frac{dt}{\ln (x)t-1}=\frac{dx}{x}\\$
still cant beak it as one type of variable on each side
so i substitute by another variable
$z=\ln x\\$
$dz=\frac{dx}{x}\\$
$\frac{dt}{zt-1}=dz\\$
i dont know how to separate each variable type on one side
so i could integrate

??

**songoku** · August 16th 2009, 05:53 PM

Hi transgalactic

Do you know integrating factor ?

**transgalactic** · August 16th 2009, 09:05 PM

yes i know that.but how did you know that that i should solve it like this in here?

**Chris L T521** · August 16th 2009, 09:12 PM

Originally Posted by transgalactic

[snip]

$\frac{xy'}{(\ln x\arctan y)-1}=(1+y^2)\arctan y\\$
$t=\arctan y\\$
$t'=\frac{1}{1+y^2}y'\\$
$\frac{x}{\ln (x)t-1}=\frac{t}{t'}\\$

$\frac{x}{\ln (x)t-1}=\frac{tdx}{dt}\\$
$xdt=(\ln (x)t-1)tdx\\$
$\frac{dt}{\ln (x)t-1}=\frac{dx}{x}\\$

[/snip]

From here,

$\frac{\,dt}{t\ln x-1}=\frac{\,dx}{x}\implies \frac{\,dt}{\,dx}=\frac{\ln x}{x}t-\frac{1}{x}$ $\implies \frac{\,dt}{\,dx}-\frac{\ln x}{x}t=-\frac{1}{x}\sim \frac{\,dt}{\,dx}+P(x)t=Q(x)$ .

Can you take it from here and apply the integrating factor technique (as songoku mentioned)?

**transgalactic** · August 16th 2009, 10:14 PM

ok i rearanged it like i used to in order to find integration factor
$xdt=(tlnx-1)dx$
i got that the integration factor is:
$y=e^{\frac{ln^2x}{2}-lnx}$
so i mltiply both sides by y
$xe^{\frac{ln^2x}{2}-lnx}dt=(tlnx-1)e^{\frac{ln^2x}{2}-lnx}dx$
now its even more complicated than before
??

**songoku** · August 16th 2009, 11:14 PM

Hi transgalactic

Sorry, I have tried to solve this and ended up finding the integration that can't be stated in elementary function.

For now, maybe we can wait for Chris L T521's help

**transgalactic** · August 16th 2009, 11:46 PM

this is a fll solution:
http://i30.tinypic.com/2lllrpk.gif

its really frustrating that i did all the things by the book and i didnt manage to solve its
what is the mistake in diagnozing the wa of solution

what should step there in my equation is a red light to solve it like
the solution above?

**songoku** · August 16th 2009, 11:54 PM

Hi transgalactic

I re-check your work and find out the mistake.

$xdt=(\ln (x)t-1)tdx\\$

$\frac{dt}{\ln (x)t-1}=\frac{dx}{x}\\$ -----------> you missed the term "t"

So, that's not linear form anymore but Bernoulli form

**transgalactic** · August 17th 2009, 12:21 AM

$ \frac{dt}{(\ln (x)t-1)t}=\frac{dx}{x}\\ $
$ z=\ln x $
$ \frac{dt}{(zt-1)t}=dz\\ $

i dont have here a bernuly equation ..
what to do?
bernule needs to have f'() form
i dont know how transform it to this form

**songoku** · August 17th 2009, 12:31 AM

Hi transgalactic

$\frac{dt}{(zt-1)t}=dz\\$

$\frac{dt}{dz}=zt^2-t$

$\frac{dt}{dz}+t=zt^2$

**transgalactic** · August 17th 2009, 12:51 AM

here is another problem:
all those variable are confusing
how did you know to t it in this way
dt/dz

or

dz/dt

why you put the derivative part in that way

what is the logic behind it
?

**songoku** · August 17th 2009, 01:06 AM

Hi transgalactic

It's not matter whether you solve $\frac{dt}{dz}$ or $\frac{dz}{dt}$ because it will give you the same result.

If you state it in $\frac{dz}{dt}$ , it will be :

$\frac{dz}{dt}=\frac{1}{zt^2-t}$

Can you solve it ?

**transgalactic** · August 17th 2009, 01:08 AM

the problem is that i cant see what variable is a function of what variable
if i had dy and dx
then i whould do
dy/dx

but here dz and dt
i dont know who is more superior

**Chris L T521** · August 17th 2009, 01:12 PM

Originally Posted by Chris L T521

From here,

$\frac{\,dt}{t\ln x-1}=\frac{\,dx}{x}\implies \frac{\,dt}{\,dx}=\frac{\ln x}{x}t-\frac{1}{x}$ $\implies \frac{\,dt}{\,dx}-\frac{\ln x}{x}t=-\frac{1}{x}\sim \frac{\,dt}{\,dx}+P(x)t=Q(x)$ .

Can you take it from here and apply the integrating factor technique (as songoku mentioned)?

The integrating factor is $\varrho\left(x\right)=e^{-\int\frac{\ln x}{x}\,dx}=e^{-\frac{\ln^2\left(x\right)}{2}}$ .

So it follows now that $\frac{\,d}{\,dx}\left[te^{-\frac{\ln^2\left(x\right)}{2}}\right]=-\frac{1}{x}e^{\frac{-\ln^2\left(x\right)}{2}}$

Thus, $te^{-\frac{\ln^2\left(x\right)}{2}}=-\int\frac{1}{x}e^{\frac{-\ln^2\left(x\right)}{2}}\,dx\implies t=-e^{\frac{\ln^2\left(x\right)}{2}}\cdot\int\frac{1} {x}e^{\frac{-\ln^2\left(x\right)}{2}}\,dx$ .

Therefore, $\arctan y=-e^{\frac{\ln^2\left(x\right)}{2}}\cdot\int\frac{1} {x}e^{\frac{-\ln^2\left(x\right)}{2}}\,dx\implies y=\tan\left[-e^{\frac{\ln^2\left(x\right)}{2}}\cdot\int\frac{1} {x}e^{\frac{-\ln^2\left(x\right)}{2}}\,dx\right]$

It looks weird, but I guess that's the solution you're looking for...

**songoku** · August 17th 2009, 08:13 PM

Hi Chris L T521 and transgalactic

@ Chris L T521:
There is error in OP's work.

@transgalactic :
Example : $\frac{dy}{dx}=x$

$dy=x\; dx$

$y=\frac{1}{2}\; x^2 + c$

If you re-arrange the question : $\frac{dx}{dy}=\frac{1}{x}$

$x\; dx = dy$

$\frac{1}{2}x^2+c=y$

You will have same result. So, it doesn't matter if you solve $\frac{dt}{dz}$ or $\frac{dz}{dt}$

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