设函数f（x,y）可微,f对x偏导=-f（x,y）,f（0,pi/2）=1,且满足lim(n趋向于无穷)（f（0,y+(1/n)）/f(0,y)）的n次方=e的coty次方,求f(x,y)

Question

设函数f（x,y）可微,f对x偏导=-f（x,y）,f（0,pi/2）=1,且满足lim(n趋向于无穷)
（f（0,y+(1/n)）/f(0,y)）的n次方=e的coty次方,求f(x,y)

Answer 1

【e^x*f(x,y)】对x的偏导数＝e^x【f对x偏导+f(x,y)】=0,因此e^x*f(x,y)与x无关,设为g(y),即
e^xf(x,y)=g(y).由条件有g(pi/2)=1.
第二个条件是f对y偏导(0,y)/f(0,y)=coty,或者【lnf(0,y)】对y偏导＝【ln|siny|】’,
于是f(0,y)=Csiny,由f(0,pi/2)=1得C＝1.即f(0,y)=siny.
再由f(x,y)=e^(－x)g(y)得g(y)=f(0,y)=siny,于是
f(x,y)=e^(－x)siny.