设z=f(x,y)二阶连续可偏导,且=x+1,f’x(x,0)=2x,f(0,y)=sin2y,则f(x,y)=________.

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问题 设z=f(x,y)二阶连续可偏导,且=x+1,f’x(x,0)=2x,f(0,y)=sin2y,则f(x,y)=________.

选项

答案[*]x2y+xy+x2+sin2y

解析=x+1,得=xy+y+φ(x),
由f’x(x,0)=2x,得φ(x)=2x,即=xy+y+2x;
=xy+y+2x得z=x2y+xy+x2+h(y),
再由f(0,y)=sin2y得h(y)=sin2y,故f(x,y)=x2y+xy+x2+sin2y.
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