Find the derivative $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$ of the following real function $y(x)$ defined for $0 < x < 1$: $$y(x) = (\arccos x)^{\log x}$$ where $0 < \arccos x < \pi/2$.
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Derive the simultaneous ordinary differential equations for complex-valued functions $f(x)$ and $g(x)$, based on the change of variables $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$. Show that the value of $|f(x)|^{2} + |g(x)|^{2}$ is independent of $x$, where $|A|$ denotes the absolute value of a complex number $A$.
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$. Let $a = 0.8$ and $b = 0.6$. Solve the simultaneous ordinary differential equations derived in Question II.1 using the initial values $f(0) = 1$ and $g(0) = 0$, and obtain $f(x)$ and $g(x)$.