Question

Дифференциальное уравнение
x(y²+1)dx = -y(1+x²)dx

Answer 1

Ответ:

$\displaystyle x\, (y^2+1)\, dx=-y(1+x^2)\, dy\\\\\\\int \frac{x\, dx}{1+x^2}=-\int \frac{y\, dy}{y^2+1}\\\\\\\frac{1}{2}\int \frac{2x\, dx}{1+x^2}=-\frac{1}{2}\int \frac{2y\, dy}{y^2+1}\\\\\\\frac{1}{2}\, ln(1+x^2)=-\frac{1}{2}\, ln(y^2+1)+\frac{1}{2}\, lnC\\\\\\ln(x^2+1)+ln(y^2+1)=lnC\\\\\\\boxed{\ (x^2+1)(y^2+1)=C\ }$