NCERT Class 12 · Ch 9 Exercise 9.2 · Q6
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.2 Question 6
Verify y=x sin x for xy′=y+x√(x²−y²), with printed restrictions x≠0 and (x>y or x<−y).
💡 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: The identity holds only on branches with x cos x≥0; the printed restrictions alone are insufficient.
- For y=x sin x, y′=sin x+x cos x; therefore xy′−y=x²cos x.
- The right radical term is x√(x²cos²x)=x|x cos x|. Equality requires x cos x≥0.
- For example x=2 satisfies x>y=2sin 2 but cos 2<0. The two sides then have opposite signs after subtraction of y.
- On any branch with x cos x≥0 the identity does verify. The source claim needs this extra branch condition.
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