I'm having some trouble with these, could you lend me some help?
This one for example:
y = x.y' + y'.ln(y')
I tried p = y' but all i got was this
dx + (x+ln(p))dp = 0
and then I finish with an impossible integration
I'm having some trouble with these, could you lend me some help?
This one for example:
y = x.y' + y'.ln(y')
I tried p = y' but all i got was this
dx + (x+ln(p))dp = 0
and then I finish with an impossible integration
The DE requires a quite particular approach. We have...
$\displaystyle y=x\ y^{'} + y^{'}\ \ln y^{'} $ (1)
Now if we differentiate both terms of (1) we obtain...
$\displaystyle y^{'}= y^{'} + (x+1+\ln y^{'})\ y^{''}$ (2)
... and the (2) is satisfied for...
$\displaystyle y^{''}=0$ (3)
...or...
$\displaystyle x+\ln y^{'}=-1$ (4)
Thesystem of (1) and (3) has solution...
$\displaystyle y=c\ x + c\ \ln c$ (5)
... and that is the general solution of (1). The system of (1) and (4) has as solution a 'particular solution' of (1) and it requires an easy procedure...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Your replay has been useful because I discovered an error. It is...
$\displaystyle \frac{d}{dx} (y^{'}\ \ln y^{'}) = (1+\ln y^{'})\ y^{''}$ (1)
... so that the second equation is...
$\displaystyle x+\ln y^{'}=-1$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$