设函数f(x,y)可微,=-f(x,y)(0,π/2)=1,且}=ecoty,求f(x,y).

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问题 设函数f(x,y)可微,=-f(x,y)(0,π/2)=1,且}=ecoty,求f(x,y).

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答案[*] 解得f(0,y)=Csiny. 由f(0,π/2)=1,得C=1,即f(0,y)=siny. 又由[*]=-f(x,y),得lnf(x,y)=-x+lnφ(y), 即f(x,y)=φ(y)e-x,由f(0,y)=siny,得φ(y)=siny,所以f(x,y)=e-xsiny.

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