用积分因子法解(x^2+y^2+y)dx-xdy=0
问题描述：
用积分因子法解(x^2+y^2+y)dx-xdy=0
答
∵(x^2+y^2+y)dx-xdy=0
==>(x^2+y^2)dx+ydx-xdy=0
==>dx+(ydx-xdy)/(x^2+y^2)=0 (等式两端同乘积分因子1/(x^2+y^2))
==>dx+[(ydx-xdy)/y^2]/[(x/y)^2+1]=0 (分式分子分母同除y^2)
==>dx+d(x/y)/[(x/y)^2+1]=0
==>x+arctan(x/y)=C (C是任意常数)
∴原方程的通解是x+arctan(x/y)=C.