Question

设函数f(x)与g(x)可导,且有f'(x)=g(x),g'(x)=f(x),f(0)=0,g(x

设函数f(x)与g(x)可导,且有f'(x)=g(x),g'(x)=f(x),f(0)=0,g(x)不等于0,则函数F(x)=f(x)／g(x)=

Answer 1

f'(x)=g(x) ，g'(x)=f(x)
G(x)=f(x)+g(x)
G'(x)=f'(x)+g'(x)=g(x)+f(x)=G(x)
G(x)=c₁e^x ①
H(x)=f(x)-g(x)
H'(x)=f'(x)-g'(x)=g(x)-f(x)=-H(x)
H(x)=c₂e^-x ②
∴f(x)=[G(x)+H(x)]/2=c₁e^x +c₂e^-x
g(x)=f'(x)=c₁e^x-c₂e^-x
f(0)=c₁+c₂=0→f(x)=c₁(e^x-e^-x)
g(x)=c₁(e^x+e^-x)
∴F(x)=f(x)/g(x)=(e^x-e^-x)/(e^x+e^-x)