Question

Решить уравнение (x+y)dx-(y-x)dy=0

Answer 1

$(x+y)dx+(x-y)dy=0\\ (x+y)'_y=1=(x-y)'_x\\ U(x,y)=\int(x+y)dx=\dfrac{x^2}{2}+xy+\phi(y)\\ x+\phi'(y)=x-y=>\phi'(y)=-y=>\phi(y)=C_1-\dfrac{y^2}{2}\\ \dfrac{x^2}{2}+xy-\dfrac{y^2}{2}=C_2\\ (y-x)^2=2x^2+C\\ y=x\pm\sqrt{2x^2+C}$