NCERT Class 12 · Ch 9 Miscellaneous Exercise · Q4
NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 4
Solve y′+√[(1−y²)/(1−x²)]=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: For |x|<1, |y|<1: sin⁻¹y+sin⁻¹x=C, plus y=±1. For |x|>1, |y|>1: sign(y)acosh|y|+sign(x)acosh|x|=C, plus y=±1.
- The original denominator excludes x=±1. A real radical requires |y|≤1 for |x|<1, or |y|≥1 for |x|>1.
- On the interior component, separate dy/√(1−y²)=−dx/√(1−x²) and integrate to obtain sin⁻¹y+sin⁻¹x=C. Restrict the resulting branch to y′≤0.
- On an exterior component use √[(y²−1)/(x²−1)]. The primitive of 1/√(u²−1) is sign(u)acosh|u|, yielding the exterior relation.
- Division by the radical loses y=±1, which satisfy the original equation for every x≠±1. Piecewise continuations may meet these equilibrium levels at non-Lipschitz endpoints.
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