NCERT Class 12 · Ch 9 Exercise 9.5 · Q6
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.5 Question 6
Solve y′+\(\frac{2}{x}\) y=\(x \log{\left(x \right)}\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: y = \(\frac{16 C + 4 x^{4} \log{\left(x \right)} - x^{4}}{16 x^{2}}\), for x>0.
- Write y′ + \(\frac{2}{x}\) y = \(x \log{\left(x \right)}\); work for x>0.
- An integrating factor is μ=\(x^{2}\), since μ′/μ=\(\frac{2}{x}\).
- Multiply through: d(μy)/dx = \(x^{3} \log{\left(x \right)}\).
- Integrate the right side: \(\frac{x^{4} \log{\left(x \right)}}{4} - \frac{x^{4}}{16}\). Thus μy=\(\frac{x^{4} \log{\left(x \right)}}{4} - \frac{x^{4}}{16}\)+C.
- Therefore y = \(\frac{16 C + 4 x^{4} \log{\left(x \right)} - x^{4}}{16 x^{2}}\).
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