NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 11

✅ Step-by-Step Solution

1. Write y′ + \(\cot{\left(x \right)}\) y = \(4 x \csc{\left(x \right)}\); work for 0<x<π.
2. An integrating factor is μ=\(\sin{\left(x \right)}\), since μ′/μ=\(\frac{1}{\tan{\left(x \right)}}\).
3. Multiply through: d(μy)/dx = \(4 x\).
4. Integrate the right side: \(2 x^{2}\). Thus μy=\(2 x^{2}\)+C.
5. Substitute x=\(\frac{\pi}{2}\), y=\(0\): C=\(- \frac{\pi^{2}}{2}\).
6. Therefore y = \(\frac{\left(2 x - \pi\right) \left(2 x + \pi\right)}{2 \sin{\left(x \right)}}\).