Question

Помогите решить диф.уравн.
xy(1+x^2)*y'=1+y^2

Answer 1

$xy(1+x^2)cdot y'=1+y^2\\xy(1+x^2)cdot frac{dy}{dx}=1+y^2\\ int frac{y, dy}{1+y^2} = int frac{dx}{x(1+x^2)} \\.quad frac{1}{x(1+x^2)} = frac{A}{x} + frac{Bx+C}{1+x^2} = frac{A(1+x^2)+x(Bx+C)}{y} \\x^2; |; A+B=0\\x; |; C=0\\x^0; |; A=1quad Rightarrow quad B=-A=-1\\int frac{dx}{x(1+x^2)} =int frac{dx}{x}-int frac{x, dx}{1+x^2}=ln|x|-frac{1}{2}int frac{2x, dx}{1+x^2} =$

$=ln|x|-frac{1}{2}ln|1+x^2|+C_1$

$int frac{y, dy}{1+y^2}=frac{1}{2}ln|1+y^2|+C_2$

$frac{1}{2}ln(1+y^2)=lnx-frac{1}{2}ln(1+x^2)-frac{1}{2}lnC\\frac{1}{2}(ln(1+y^2)+ln(1+x^2)+lnC)=lnx\\2lnx=ln(C(1+x^2)(1+y^2)\\x^2=C(1+x^2)(1+y^2)$