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设du(x,y)=(x+2y)dx+(2x+y)dy,求u(x,y)

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设du(x,y)=(x+2y)dx+(2x+y)dy,求u(x,y)
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答案和解析
分项组合
du(x,y)=(x+2y)dx+(2x+y)dy
=xdx+ydy+2(xdy+ydx)
=d(1/2×x^2)+d(1/2×y^2)+2d(xy)
=d(1/2×x^2+1/2×y^2+2xy)
所以,u(x,y)=1/2×x^2+1/2×y^2+2xy+C