Question

2y'= y^2/x^2+6*y/x+6 помогите пожалуйста решить

Answer 1

$2y'=(\frac{y}{x})^2+6\cdot \frac{y}{x}+6\\\\u=\frac{y}{x},\; y=ux,\; y'=u'x+u\\\\2(u'x+u)=u^2+6u+6\\\\2u'x=u^2+4u+6\\\\u'=\frac{du}{dx}=\frac{u^2+4u+6}{2x}\\\\\int \frac{du}{u^2+4u+6}=\int \frac{dx}{2x}\\\\\int \frac{du}{(u+2)^2+2}=\frac{1}{2}\int \frac{dx}{x}\\\\\frac{1}{\sqrt2}\cdot arctg\frac{u+2}{\sqrt2}=\frac{1}{2}\cdot ln|x|+C$

$\sqrt2\cdot arctg\frac{\frac{y}{x}+2}{\sqrt2}=ln|x|+C\\\\\sqrt2\cdot arctg\frac{y+2x}{x\sqrt2}=\ln|x|+C$