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We are given the differential equation $\frac{dy}{dx} = y \cot 2x$. To solve this, we first separate the variables $y$ and $x$ to get: $$\frac{dy}{y} = \cot 2x ~dx$$
Now, we integrate both sides of the equation: $$\int \frac{dy}{y} = \int \cot 2x ~dx$$ The integral of $\frac{1}{y}$ with respect to $y$ is $\ln |y|$. The integral of $\cot 2x$ with respect to $x$ is $\frac{1}{2} \ln |\sin 2x|$. Thus, we have: $$\ln |y| = \frac{1}{2} \ln |\sin 2x| + C$$ where $C$ is the constant of integration.
We can rewrite the equation as: $$\ln |y| = \ln |\sqrt{\sin 2x}| + C$$ Taking the exponential of both sides, we get: $$|y| = e^{\ln |\sqrt{\sin 2x}| + C} = e^{\ln |\sqrt{\sin 2x}|} \cdot e^C$$ $$y = A \sqrt{\sin 2x}$$ where $A = \pm e^C$ is another constant.
We are given that $y(\frac{\pi}{4}) = 2$. Substituting $x = \frac{\pi}{4}$ and $y = 2$ into the general solution, we get: $$2 = A \sqrt{\sin (2 \cdot \frac{\pi}{4})} = A \sqrt{\sin (\frac{\pi}{2})} = A \sqrt{1} = A$$ Thus, $A = 2$.
Substituting $A = 2$ into the general solution, we obtain the particular solution: $$y = 2 \sqrt{\sin 2x}$$
Final Answer: $y = 2\sqrt{\sin 2x}$
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