NCERT Class 12 · Ch 9 Miscellaneous Exercise · Q10
NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 10
Solve [e^(−2√x)/√x−y/√x] dx/dy=1, x≠0. The real equation requires x>0.
💡 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 = \(\left(C + 2 \sqrt{x}\right) e^{- 2 \sqrt{x}}\), for x>0.
- Write y′ + \(\frac{1}{\sqrt{x}}\) y = \(\frac{e^{- 2 \sqrt{x}}}{\sqrt{x}}\); work for x>0.
- An integrating factor is μ=\(e^{2 \sqrt{x}}\), since μ′/μ=\(\frac{1}{\sqrt{x}}\).
- Multiply through: d(μy)/dx = \(\frac{1}{\sqrt{x}}\).
- Integrate the right side: \(2 \sqrt{x}\). Thus μy=\(2 \sqrt{x}\)+C.
- Therefore y = \(\left(C + 2 \sqrt{x}\right) e^{- 2 \sqrt{x}}\).
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