求通解dy/dx=sin(x-y)

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求通解dy/dx=sin(x-y)
求通解dy/dx=sin(x-y)

求通解dy/dx=sin(x-y)
∵dy/dx=sin(x-y)
==>dy=sin(x-y)dx
==>dx-dy=dx-sin(x-y)dx
==>d(x-y)=(1-sin(x-y))dx
==>d(x-y)/(1-sin(x-y))=dx
==>d(x-y)/(sin((x-y)/2)-cos((x-y)/2))^2=dx (应用(sin((x-y)/2)^2+(cos((x-y)/2)^2=1)
==>(sec((x-y)/2))^2d(x-y)/(tan((x-y)/2)-1)^2=dx (分子分母同除(cos((x-y)/2))^2)
==>2d(tan((x-y)/2)-1)/(tan((x-y)/2)-1)^2=dx
==>2/(1-tan((x-y)/2))=x+C (C是常数)
==>(x+C)(1-tan((x-y)/2))=2
∴原方程的通解是(x+C)(1-tan((x-y)/2))=2.

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