Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 1

For real $x>0$, use logarithmic differentiation: $\log y=\tan x\log x$. The product rule gives $y^{\prime }/y={\sec }^{2}x\log x+\tan x/x$, henceWork on an interval where $\tan x$ is defined. The positive-base condition makes the variable real power unambiguous.

Implicit differentiation gives $2y^{\prime }=15x^2+(\cos y)y^{\prime }$. ThereforeFor real $y$, the denominator is at least one, so division introduces no singularity. In fact the function $2y-\sin y$ is strictly increasing, consistent with a unique real branch $y(x)$.