求微分方程通解用适当的变量代换求下列微分方程的通解：y'=cosxcosy+sinxsiny谢谢我不会做求高人指点!

Question

求微分方程通解
用适当的变量代换求下列微分方程的通解：y'=cosxcosy+sinxsiny
谢谢 我不会做 求高人指点!

Answer 1

y' = cosxcosy + sinxsiny = cos(x-y)
令x-y = t,则dt = dx-dy
又y' = dy/dx,代入原式得
dy/dx = cost,因dt = dx-dy
则dt = dx - costdx
dx = dt/(1-cost) = d(t/2)/sin²(t/2)
两边积分x = -cos(t/2)/sin(t/2) + C
即x = -cos[(x-y)/2]/sin[(x-y)/2] + C
这是原方程通解