NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 14

✅ Step-by-Step Solution

1. Write y′ + \(\frac{2 x}{x^{2} + 1}\) y = \(\frac{1}{\left(x^{2} + 1\right)^{\frac{3}{2}}}\); work on intervals where the original equation is defined.
2. An integrating factor is μ=\(x^{2} + 1\), since μ′/μ=\(\frac{2 x}{x^{2} + 1}\).
3. Multiply through: d(μy)/dx = \(\frac{1}{\sqrt{x^{2} + 1}}\).
4. Integrate the right side: \(\operatorname{asinh}{\left(x \right)}\). Thus μy=\(\operatorname{asinh}{\left(x \right)}\)+C.
5. Substitute x=\(1\), y=\(0\): C=\(- \log{\left(1 + \sqrt{2} \right)}\).
6. Therefore y = \(\frac{\operatorname{asinh}{\left(x \right)} - \log{\left(1 + \sqrt{2} \right)}}{x^{2} + 1}\).