Question

Решить уравнение (x+y)dx-(y-x)dy=0

Answer 1

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