2 solutions to 函数f(x) huxlnn

calala

已知实函数f(x)满足:f(x+y)+f(x-y)=2f(x)*cosy,求f(x)

Solution 1:

f '(y)-f '(-y)=-2f(0)siny
f '(y)+f '(-y)=2f '(0)cosy
 
f '(y)=-f(0)siny+f '(0)cosy 
 
f(y)=f(0)cosy+f '(0)siny + constwww.ddhw.com

The const should be 0 to satisfy f(x+y)+f(x-y)=2f(x)*cosy.
 
So f(x) = Asin(x+x0) where A and x0 are any real numbers

Solution 2

Any function f(x) can be written as
f(x) = F(x) + G(x)
where F(-x) = F(x) and G(-x) = -G(x).

So
1) f(y)+f(-y) = 2f(0)cosy
or
F(x) = f0cosx, where f0 = f(0).

2) f(2x) + f0 = 2f(x)cosx
f0cos(2x) + G(2x) + f0 = 2f0(cosx)^2 + 2G(x)cosx
G(2x) = 2G(x)cosx
so
G(x) = f1sinx where f1 is constant.

So f(x) = f0cosx + f1sinx = Asin(x+x0)
where A and x0 are any real constantwww.ddhw.com

 

huxlnn

  结果正确