NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 9

✅ Step-by-Step Solution

1. Write y′ + \(\cot{\left(x \right)} + \frac{1}{x}\) y = \(1\); work where x≠0 and sin x≠0.
2. An integrating factor is μ=\(x \sin{\left(x \right)}\), since μ′/μ=\(\frac{x + \tan{\left(x \right)}}{x \tan{\left(x \right)}}\).
3. Multiply through: d(μy)/dx = \(x \sin{\left(x \right)}\).
4. Integrate the right side: \(- x \cos{\left(x \right)} + \sin{\left(x \right)}\). Thus μy=\(- x \cos{\left(x \right)} + \sin{\left(x \right)}\)+C.
5. Therefore y = \(- \frac{- C \tan{\left(x \right)} + x \sin{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}}{x \sin{\left(x \right)} \tan{\left(x \right)}}\).