若f(x)f(y)对任意x,y可微,且f(x+y)= f(x)f(y)当f(x)不等0 且 f(y)不等0 时,证明f(x)= e^ax,其中a为实数
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![若f(x)f(y)对任意x,y可微,且f(x+y)= f(x)f(y)当f(x)不等0 且 f(y)不等0 时,证明f(x)= e^ax,其中a为实数](/uploads/image/z/10759118-14-8.jpg?t=%E8%8B%A5f%EF%BC%88x%EF%BC%89f%EF%BC%88y%EF%BC%89%E5%AF%B9%E4%BB%BB%E6%84%8Fx%2Cy%E5%8F%AF%E5%BE%AE%2C%E4%B8%94f%EF%BC%88x%2By%EF%BC%89%3D+f%EF%BC%88x%EF%BC%89f%EF%BC%88y%EF%BC%89%E5%BD%93f%EF%BC%88x%29%E4%B8%8D%E7%AD%890+%E4%B8%94+f%EF%BC%88y%EF%BC%89%E4%B8%8D%E7%AD%890+%E6%97%B6%2C%E8%AF%81%E6%98%8Ef%EF%BC%88x%EF%BC%89%3D+e%5Eax%2C%E5%85%B6%E4%B8%ADa%E4%B8%BA%E5%AE%9E%E6%95%B0)
若f(x)f(y)对任意x,y可微,且f(x+y)= f(x)f(y)当f(x)不等0 且 f(y)不等0 时,证明f(x)= e^ax,其中a为实数
若f(x)f(y)对任意x,y可微,且f(x+y)= f(x)f(y)
当f(x)不等0 且 f(y)不等0 时,证明f(x)= e^ax,其中a为实数
若f(x)f(y)对任意x,y可微,且f(x+y)= f(x)f(y)当f(x)不等0 且 f(y)不等0 时,证明f(x)= e^ax,其中a为实数
令y=0
得f(x)=f(x)f(0)
当f(x)不等于0时则f(0)=1
f(x+y)-f(x)=f(x)(f(y)-1)
=>[f(x+y)-f(x)]/y=f(x)(f(y)-f(0))/y
设y趋近于0
则左边[f(x+y)-f(x)]/y=f'(x)
右边f(x)(f(y)-f(0))/y=f(x)f'(0)
则f'(x)=f(x)f'(0)
设f'(0)=a,(若a=0则f(x)=C,f(x+y)= f(x)f(y)可得C=1,C=0舍去)
a不等于0时
则f(x)=f'(x)/a
设f(x)=e^ax*g(x),
则e^ax*g(x)=[ae^ax*g(x)+e^ax*g'(x)]/a
=>e^ax*g'(x)/a=0
则g(x)=C
f(x)=Ce^ax
f(x+y)= f(x)f(y)
=>Ce^(ax+ay)=C²e^(ax+ay)
则C=1,C=0舍去
故f(x)=e^ax,
因a=0时f(x)=1同样成立
故f(x)=e^ax,
证明:令y=0,f(x+0)=f(x)f(0)
因为f(x)≠0,所以f(0)=1
f'(x)=lim(y→0)[f(x+y)-f(x)]/y
=lim(y→0)[f(x)f(y)-f(x)]/y
=f(x)lim(y→0)[f(y)-1]/y
=f(x)lim(y→0)[f(y)-f(0)]/y
=f(x)f'(0)
设f'(0)=a,则dy/dx=ay解该微分方程y=e^ax
完毕!