NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 4

✅ Step-by-Step Solution

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