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: Variable Separation. Rewrite the equation as: \[ \frac{1}{y} \, dy = \cot 2x \, dx \]

Step 2: Integration. Integrate both sides: \[ \int \frac{1}{y} \, dy = \int \cot 2x \, dx \] This yields: \[ \log |y| = \frac{1}{2} \log |\sin 2x| + \log c \] where \( \log c \) is the integration constant. 

Step 3: Expression Simplification. Combine logarithmic terms: \[ \log |y| = \log \left( c \sqrt{\sin 2x} \right) \] Exponentiate to remove the logarithm: \[ y = c \sqrt{\sin 2x} \] 

Step 4: Particular Solution Determination. Given \( y\left( \frac{\pi}{4} \right) = 2 \), substitute \( x = \frac{\pi}{4} \) and \( y = 2 \): \[ 2 = c \sqrt{\sin\left( 2 \cdot \frac{\pi}{4} \right)} \] This simplifies to: \[ 2 = c \sqrt{\sin\left( \frac{\pi}{2} \right)} \] Since \( \sin\left( \frac{\pi}{2} \right) = 1 \): \[ 2 = c \cdot 1 \quad \Rightarrow \quad c = 2 \] 

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

Final Answer: \[ \boxed{y = 2 \sqrt{\sin 2x}} \] This is the derived particular solution.