证明:若f(x)在负无穷大到正无穷满足f(x)的导数=f(x)且f(0)=1,证明f(x)=e的x次方

Question

证明:若f(x)在负无穷大到正无穷满足f(x)的导数=f(x)且f(0)=1,证明f(x)=e的x次方

Answer 1

f'(x)=f(x)
f'(x)/f(x)=1
(ln|f(x)|)'=1
两边积分：ln|f(x)|=x+C
令x=0得：0=0+C,C=0
所以ln|f(x)|=x
f(x)=±e^x
而f(0)=1
所以f(x)=e^x