Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #59
Solve (x² − 1)y′ + 2xy = 2/(x² − 1), where x < −1 or x > 1.
✅ Step-by-Step Solution
Final Answer: (x² − 1)y = ln|(x − 1)/(x + 1)| + C.
- After division, P = 2x/(x² − 1); a convenient IF is x² − 1 on either given interval.
- Then [(x² − 1)y]′ = 2/(x² − 1).
- Use 2/(x² − 1) = 1/(x − 1) − 1/(x + 1), whose integral is ln|(x − 1)/(x + 1)|.
- Thus (x² − 1)y = ln|(x − 1)/(x + 1)| + C, on a connected interval in the stated domain.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] After division, P = 2x/(x² − 1); a convenient IF is x² − 1 on either given interval.
- [1 mark] Then [(x² − 1)y]′ = 2/(x² − 1).
- [1 mark] Use 2/(x² − 1) = 1/(x − 1) − 1/(x + 1), whose integral is ln|(x − 1)/(x + 1)|.
- [0.5 marks] Thus (x² − 1)y = ln|(x − 1)/(x + 1)| + C, on a connected interval in the stated domain.
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