NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 13

✅ Step-by-Step Solution

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