NCERT Class 12 · Ch 9 Miscellaneous Exercise · Q7
NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 7
Solve (1+e^(2x))dy+(1+y²)eˣdx=0, 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: y=e^(−x).
- Separate dy/(1+y²)=−eˣdx/(1+e^(2x)).
- Integrate: tan⁻¹y+tan⁻¹(eˣ)=C. The initial point gives C=π/2.
- Hence y=cot(tan⁻¹eˣ)=e^(−x), a global real solution.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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