NCERT Class 12 · Ch 9 Exercise 9.3 · Q11
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.3 Question 11
Solve (x³+x²+x+1)y′=2x²+x, 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: \(\frac{\log{\left(x + 1 \right)}}{2} + \frac{3 \log{\left(x^{2} + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\), x>−1.
- Factor the denominator as (x+1)(x²+1). Partial fractions give 1/[2(x+1)]+(3x−1)/[2(x²+1)].
- Integrate: y=(1/2)log|x+1|+(3/4)log(x²+1)−(1/2)tan⁻¹x+C.
- At x=0, y=1 gives C=1; the initial-value interval is x>−1.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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