Question

6ylnx+xy'-6ylny=7y y(1)=e^-3

Answer 1

$displaystyle 6ylnx+xy'-6ylny=7y\-6ylnfrac{y}{x}+xy'=7y\y=tx;y'=t'x+t\t'x=6t+6tlnt\6intfrac{dx}{x}=intfrac{d(1+lnt)}{1+lnt}\ln|1+lnt|=6ln|x|+C\1+lny-lnx=Cx^6\1-3=Cto C=-2\1+lnfrac{y}{x}+2x^6=0$