Answer to Question #169194 in Civil and Environmental Engineering for mae

Question

Solve the homogenous equation of differential equation (x+y-2)dx+(2x-3y+1)dy=0

Answer 1

$x+y-2+\left(2x-3y+1\right)\frac{dy}{dx}=0$

$x+y-2+\left(2x-3y+1\right)y'\:=0$

$x+y-2+\left(2x-3y+1\right)y'\:-\left(x+y\right)=0-\left(x+y\right)$

$-2+\left(2x-3y+1\right)y'\:=-\left(x+y\right)$

$-2+\left(2x-3y+1\right)y'\:+2=-\left(x+y\right)+2$

$\left(2x-3y+1\right)y'\:=-x-y+2$

$\frac{\left(2x-3y+1\right)y'\:}{2x-3y+1}=-\frac{x}{2x-3y+1}-\frac{y}{2x-3y+1}+\frac{2}{2x-3y+1}$

$y'\:=\frac{-x-y+2}{2x-3y+1}$

$\left(\frac{-x+2-2xv-v}{-3v+1}\right)'\:=\frac{-x-\frac{-x+2-2xv-v}{-3v+1}+2}{2x-3\cdot \frac{-x+2-2xv-v}{-3v+1}+1}$

$\left(\frac{-x+2-2xv-v}{-3v+1}\right)'\:=v$

$\left(\frac{-x+2-2xv-v}{-3v+1}\right)'\:$

$=\frac{\left(-x+2-2xv-v\right)'\left(-3v+1\right)-\left(-3v+1\right)'\left(-x+2-2xv-v\right)}{\left(-3v+1\right)^2}$

$\frac{-5xv'\:+5v'\:+6v^2+v-1}{\left(-3v+1\right)^2}=v$

$\frac{1}{9v^3-12v^2+1}v'\:=\frac{1}{-5x+5}$

$-\frac{1}{5}\ln \left(3v-1\right)+\frac{2}{5}\left(\frac{1}{4}\ln \left(36v^2-36v-12\right)+\frac{1}{4\sqrt{21}}\left(\ln \left|\frac{\sqrt{3}\left(2v-1\right)}{\sqrt{7}}+1\right|-\ln \left|\frac{\sqrt{3}\left(2v-1\right)}{\sqrt{7}}-1\right|\right)\right)=-\frac{1}{5}\ln(-5x+5)+C_1$