求微分方程（4+x的2次方）dy=（4+y的2次方）dx的通解

Question

求微分方程（4+x的2次方）dy=（4+y的2次方）dx的通解

Answer 1

∵(4+x²)dy=(4+y²)dx ==>dy/(4+y²)=dx/(4+x²)
==>d(y/2)/(4+(y/2)²)=d(x/2)/(4+(x/2)²)
==>arctan(y/2)=arctan(x/2)+arctanC (C是积分常数)
==>y/2=tan(arctan(x/2)+arctanC)
∴y=2*tan(arctan(x/2)+arctanC)
=2*{[tan(arctan(x/2))+tan(arctanC)]/[1-tan(arctan(x/2))*tan(arctanC)]} (应用正切和角公式)
=2*[(x/2+C)/(1-Cx/2)]
=2(x+2C)/(2-Cx)
故原微分方程的通解是y=2(x+2C)/(2-Cx) (C是积分常数).