百度作业网(1-x^2)dy/dx+xy=1
来源：学生作业帮 编辑：百度作业网作业帮 分类：数学作业 时间：2024/07/07 23:34:59
(1-x^2)dy/dx+xy=1
∵(1-x^2)dy/dx+xy=1
==>(1-x^2)dy+xydx=dx
==>dy/(1-x^2)^(1/2)+xydx/(1-x^2)^(3/2)=dx/(1-x^2)^(3/2) (等式两端同除(1-x^2)^(3/2))
==>dy/(1-x^2)^(1/2)+yd(1/(1-x^2)^(1/2))=dx/(1-x^2)^(3/2)
==>d(y/(1-x^2)^(1/2))=d(x/(1-x^2)^(1/2))
==>y/(1-x^2)^(1/2)=x/(1-x^2)^(1/2)+C (C是常数)
==>y=x+C(1-x^2)^(1/2)
∴原方程的通解是y=x+C(1-x^2)^(1/2).
==>(1-x^2)dy+xydx=dx
==>dy/(1-x^2)^(1/2)+xydx/(1-x^2)^(3/2)=dx/(1-x^2)^(3/2) (等式两端同除(1-x^2)^(3/2))
==>dy/(1-x^2)^(1/2)+yd(1/(1-x^2)^(1/2))=dx/(1-x^2)^(3/2)
==>d(y/(1-x^2)^(1/2))=d(x/(1-x^2)^(1/2))
==>y/(1-x^2)^(1/2)=x/(1-x^2)^(1/2)+C (C是常数)
==>y=x+C(1-x^2)^(1/2)
∴原方程的通解是y=x+C(1-x^2)^(1/2).
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