Question

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

Answer 1

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