f continuous and f^2(x)=(4e^x)*f(x)+1 for every x in R and f(0)=2-sqrt(5).find f(x)
+6251
\(u=f(x)\)
\(u^2 - 4 e^x u - 1 = 0\)
\(\left(u - 2 e^x\right)^2 - 4e^{2x}-1=0\)
\(\left(u-2e^x\right)^2=1+4e^{2x}\)
\(u-2e^x = \pm \sqrt{1+4e^{2x}}\)
\(u=2e^x \pm \sqrt{1+4e^{2x}}\)
\(f(x) = 2e^x \pm \sqrt{1+4e^{2x}}\)
\(f(0)=2\pm \sqrt{1+4}=2-\sqrt{5} \Rightarrow f(x) = 2e^x - \sqrt{1+4e^{2x}}\)
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