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Find a Few Functions (Posted on 2018-04-15) Difficulty: 3 of 5
Find solutions for f(x) and g(x), such that:

(1): {f(x) + f'(x)}^2 = 1 + f(2x)

(2): {g(x) + g'(x)}^2 = 2*(g(x) + g(2x))

No Solution Yet Submitted by Larry    
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Some Thoughts success for part 1; not so much for part 2 | Comment 1 of 3
For part 1, the first try succeeded:

f(x)=sin(x)
f'(x)=cos(x)

(sin(x)+cos(x))^2 = sin^2(x) + cos^2(x) + 2*sin(x)*cos(x)
                  = 1 + sin(2x) = 1 + f(2x)

I tried a couple for part 2, but neither panned out:                  
                  
g(x)=cos(x)
g'(x)=-sin(x)

(cos(x) - sin(x))^2 = cos^2(x) + sin^2(x) - 2*sin(x)*cos(x)
                    = 1 - 2*sin(x)*cos(x)
                    = 1 - sin(2x) = 1 + g'(2x)
                    
g(x) = e^x
g'(x) = e^x

(e^x + e^x)^2 = 4*e^(2x)

  Posted by Charlie on 2018-04-15 12:33:53
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