NCERT Class 12 · Ch 9 Exercise 9.3 · Q13
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.3 Question 13
Solve cos(y′)=a, a∈R, with y(0)=1.
💡 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 |a|≤1: y=1+mx, m=2nπ±cos⁻¹a, n∈Z. For |a|>1: no real solution.
- A real cosine lies in [−1,1], so no real slope is possible if |a|>1.
- Otherwise the allowed slopes are the discrete values m=2nπ±cos⁻¹a. A derivative cannot jump between these values on a connected interval (Darboux property), so it is constant there.
- Integrate y=mx+C; the initial condition gives C=1.
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