NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 1

✅ Step-by-Step Solution

1. Write y′ + \(2\) y = \(\sin{\left(x \right)}\); work on intervals where the original equation is defined.
2. An integrating factor is μ=\(e^{2 x}\), since μ′/μ=\(2\).
3. Multiply through: d(μy)/dx = \(e^{2 x} \sin{\left(x \right)}\).
4. Integrate the right side: \(\frac{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x}}{5}\). Thus μy=\(\frac{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x}}{5}\)+C.
5. Therefore y = \(\frac{\left(5 C + 2 e^{2 x} \sin{\left(x \right)} - e^{2 x} \cos{\left(x \right)}\right) e^{- 2 x}}{5}\).