Question

(X-1)dy-(y^2-6y)dx=0

Answer 1

$(x-1)dy-(y^2-6y)dx=0;$

$\frac{dy}{(y^2-6y)}-\frac{dx}{x-1}=0;$

$\frac{dy}{y^2-6y} =\frac{dx}{x-1}$

$\int\frac{dy}{y^2-6y} =\int\frac{dx}{x-1} ;$

1) $\int\frac{dy}{y^2-6y}=\left \{ {{\frac{A}{y}+\frac{B}{y-6}=\frac{1}{y(y-6)} } \atop {A(y-6)+By=1}}\atop{B=\frac{1}{6} };A=-\frac{1}{6} \right.$   $\int\frac{dy}{y^2-6y}=\int-\frac{1}{6y}+\frac{1}{6(y-6)}dy=\frac{1}{6}\int\frac{1}{y-6}-\frac{1}{6}\int\frac{1}{y}= \frac{1}{6}ln|y-6|-\frac{1}{6}ln|y|=\frac{1}{6}ln|\frac{y-6}{y} |+C$

2) $\int\frac{dx}{x-1}=ln|x-1|+C$

$\frac{1}{6}ln|\frac{y-6}{y} |=ln|x-1|+C$

$ln|\frac{y-6}{y} |=6ln|xC-C|;$

$\frac{y-6}{y}=(xC-C)^{6};$

$y=\frac{6}{1-(xC-C)^6}$

$\boxed{y=\frac{6}{1-C(x-1)^6} }$