NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 5

✅ Step-by-Step Solution

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