Match trigonometric and exponential series

$f^\prime=a\sec x\tan x+b\sec^2x=a\sec x\tan x+b\tan^2x+b=(a\sec x+b\tan x)\tan x+b=f\tan x+b$.

$f^{\prime\prime}=f^\prime\tan x+f\sec^2x$. At zero, $f(0)=a$, $f^\prime(0)=b$, $f^{\prime\prime}(0)=a$. Hence $f(x)=a+bx+(a/2)x^2+O(x^3)$.

$e^{\sqrt2 x}\cos x=(1+\sqrt2 x+(\sqrt2 x)^2/2+O(x^3))(1-x^2/2+O(x^4))=1+\sqrt2 x+x^2/2+O(x^3)$. Comparing gives $a=1$ and $b=\sqrt2$, which are positive and distinct.