Question

Find the particular solution of the differential equation \( \frac{dy}{dx} = y \cot 2x \), given that \( y\left(\frac{\pi}{4}\right) = 2 \).

Hint:

For separable differential equations, isolate \( y \) and \( x \), then integrate both sides.

Answer 1

Step 1: Separating the Variables Rewriting the equation: \[ \frac{1}{y} \, dy = \cot 2x \, dx \] 

Step 2: Integrating Both Sides Integrate both sides: \[ \int \frac{1}{y} \, dy = \int \cot 2x \, dx \] The left-hand side becomes: \[ \log |y| \] The right-hand side uses the integral of \( \cot 2x \): \[ \int \cot 2x \, dx = \frac{1}{2} \log |\sin 2x| \] So the equation becomes: \[ \log |y| = \frac{1}{2} \log |\sin 2x| + \log c \] Here, \( \log c \) is the constant of integration. 

Step 3: Simplify the Expression} Combine the logarithms: \[ \log |y| = \log \left( c \sqrt{\sin 2x} \right) \] Exponentiate both sides to remove the logarithm: \[ y = c \sqrt{\sin 2x} \] 

Step 4: Finding the Particular Solution} We are given the condition \( y\left( \frac{\pi}{4} \right) = 2 \). Substitute \( x = \frac{\pi}{4} \) and \( y = 2 \) into the solution: \[ 2 = c \sqrt{\sin\left( 2 \cdot \frac{\pi}{4} \right)} \] Simplify: \[ 2 = c \sqrt{\sin\left( \frac{\pi}{2} \right)} \] Since \( \sin\left( \frac{\pi}{2} \right) = 1 \), we have: \[ 2 = c \cdot 1 \quad \Rightarrow \quad c = 2 \] 

Step 5: Final Solution} Substitute \( c = 2 \) back into the solution: \[ y = 2 \sqrt{\sin 2x} \] 

Final Answer: \[ \boxed{y = 2 \sqrt{\sin 2x}} \] This is the required particular solution to the given differential equation.