Question

Решите диф. уравнение

Answer 1

Ответ:

$y'=\dfrac{x+y}{x-y}\\\\\\u=\dfrac{y}{x}\ \ ,\ \ y=ux\ \ ,\ \ y'=u'x+u\\\\u'x+u=\dfrac{x+ux}{x-ux}\ \ ,\ \ \ u'x=\dfrac{1+u}{1-u}-u\ \ ,\ \ u'x=\dfrac{1+u^2}{1-u}\ \ ,\\\\\\\int \dfrac{(1-u)\, du}{1+u^2}=\int\dfrac{dx}{x}\ \ ,\ \ \ \int \dfrac{du}{1+u^2}-\dfrac{1}{2}\int \dfrac{2u\, du}{1+u^2}=\int \dfrac{dx}{x} \ \ ,\\\\\\arctgu-\dfrac{1}{2}\, ln|1+u^2|=ln|x|+lnC\\\\\\arctg\dfrac{y}{x}-ln\sqrt{1+\dfrac{y^2}{x^2}}=ln\, (C|x|)\\\\\\arctg\dfrac{y}{x}-ln\dfrac{\sqrt{x^2+y^2}}{|x|}=ln\, (C|x|)$