NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 7

✅ Step-by-Step Solution

1. Write y′ + \(\frac{1}{x \log{\left(x \right)}}\) y = \(\frac{2}{x^{2}}\); work for x>0, x≠1; use the undivided equation to examine any possible extension at 1.
2. An integrating factor is μ=\(\log{\left(x \right)}\), since μ′/μ=\(\frac{1}{x \log{\left(x \right)}}\).
3. Multiply through: d(μy)/dx = \(\frac{2 \log{\left(x \right)}}{x^{2}}\).
4. Integrate the right side: \(\frac{- 2 \log{\left(x \right)} - 2}{x}\). Thus μy=\(\frac{- 2 \log{\left(x \right)} - 2}{x}\)+C.
5. Therefore y = \(\frac{C x - 2 \log{\left(x \right)} - 2}{x \log{\left(x \right)}}\).
6. At x=1 the undivided equation requires y(1)=0. Only C=2 has a removable singularity in the displayed family; it extends with y(1)=0 and y′(1)=1. Other constants cannot give a finite solution there.