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: Separate the Variables
The given differential equation is: \[ \frac{dy}{dx} = y \cot 2x. \] Separating the variables: \[ \frac{dy}{y} = \cot 2x \, dx. \]
Step 2: Integrate Both Sides
Integrating the left-hand side: \[ \int \frac{dy}{y} = \ln |y|. \] Integrating the right-hand side: \[ \int \cot 2x \, dx = \frac{1}{2} \ln |\sin 2x|. \] Thus, we get: \[ \ln |y| = \frac{1}{2} \ln |\sin 2x| + C. \]
Step 3: Solve for \( y \)
Exponentiating both sides: \[ y = e^C \cdot |\sin 2x|^{1/2}. \] Let \( e^C = k \), so: \[ y = k \sqrt{|\sin 2x|}. \]
Step 4: Apply Initial Condition
Given that \( y\left(\frac{\pi}{4}\right) = 2 \), substitute \( x = \frac{\pi}{4} \): \[ 2 = k \sqrt{|\sin (\frac{\pi}{2})|}. \] Since \( \sin (\pi/2) = 1 \), we get: \[ 2 = k \cdot 1 \quad \Rightarrow \quad k = 2. \]
Step 5: Final Particular Solution
Substituting \( k = 2 \): \[ y = 2 \sqrt{|\sin 2x|}. \]
Conclusion: The particular solution is: \[ y = 2 \sqrt{|\sin 2x|}. \]