Question

证明:若函数f(x)在区间负无穷到正无穷内满足f'(x)=f(x),且f(0)=1,则f(x)=e的x次方

Answer 1

f'-f=0 两边乘以e^-x有(f*e^-x)'=0 两边积分有f*e^-x=C f=Ce^x f(0)=1所以C=1 故f=e^x

Answer 2

证明：由f'(x)=f(x),得f（x）=e~x+a
由且f(0)=1,则a=0
所以f(x)=e的x次方