Question #169194

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

Expert's answer

"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"

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