求解微分方程xy'ln(x)sin(y)+cos(y)(1-x*cos(y))=0

来源：学生作业帮编辑：搜狗做题网作业帮分类：数学作业时间：2024/06/13 16:22:53

∵xy'ln(x)sin(y)+cos(y)(1-x*cos(y))=0
==>xln(x)(cos(y))'=cos(y)(1-x*cos(y))
==>xln(x)t'=t(1-xt) (令t=cos(y))
∴xln(x)z'+z=x (令z=1/t).(1)
∵微分方程(1)是一阶线性方程,用常数变易法可求得它的通解是
z=(x+C)/ln(x) (C是积分常数)
∴cosy=t=1/z=ln(x)/(x+C)
故原微分方程的通解是cosy=t=1/z=ln(x)/(x+C) (C是积分常数).

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