Past exam of the mathematical methods I course of the University of Oxford / 2021 / 2 / Solution

Using the hyperbolic functions identities $\sin (ix)=i\sinh x$ and $\cos (ix)=\cosh x$, divide by $i$ and put $u=e^x\ne 0$. The equation becomes $-u/2-5/(2u)=3$, henceTaking all branches of the complex logarithm givesIf $x$ is required to be real, there are no real solutions, since $e^x$ cannot be negative. The complex branches are essential to the unrestricted answer.