NCERT Class 12 · Ch 9 Miscellaneous Exercise · Q5
NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 5
Show y′+(y²+y+1)/(x²+x+1)=0 has family x+y+1=A(1−x−y−2xy).
💡 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: x+y+1=A(1−x−y−2xy), plus limiting member 1−x−y−2xy=0.
- Separate dy/(y²+y+1)=−dx/(x²+x+1). Complete squares to integrate.
- This gives tan⁻¹((2y+1)/√3)+tan⁻¹((2x+1)/√3)=C.
- Taking the tangent and absorbing constants yields x+y+1=A(1−x−y−2xy).
- The finite-A form misses the limiting family member 1−x−y−2xy=0, which also solves the ODE. The inverse-tangent relation is the fuller local description.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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