(x+y)dy+(x-y)dx=0求通解

∵(x+y)dy+(x-y)dx=0==>(1+y/x)dy+(1-y/x)dx=0设y=xt,则dy=tdx+xdt∴(x+y)dy+(x-y)dx=0==>(1+t)(tdx+xdt)+(1-t)dx=0==>(t²+1)dx+x(t+1)dt=0==>dx/x+(t+1)/(t²+1)dt=0==>ln|x|+∫t/(t²+1)dt+∫1/(t²+1)dt=ln|C| (C是积分常数)==>ln|x|+1/2∫d(t²+1)/(t²+1)+arctant=ln|C|==>ln|x|+1/2ln(t²+1)+arctant=ln|C|==>x√(t²+1)=Ce^(arctant)==>√(x²+y²)=Ce^(arctant) (C是积分常数)。