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))
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)
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Posted by Charlie
on 2018-04-15 12:33:53 |
success for part 1; not so much for part 2
