Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #98
Solve (x + 1)y′ − y = e^(3x)(x + 1)³.
✅ Step-by-Step Solution
Final Answer: y/(x + 1) = e^(3x)(3x + 2)/9 + C, for x ≠ −1.
- Divide by x + 1. P = −1/(x + 1), so choose IF 1/(x + 1).
- Then [y/(x + 1)]′ = e^(3x)(x + 1).
- By parts, ∫e^(3x)(x + 1)dx = e^(3x)[(x + 1)/3 − 1/9] = e^(3x)(3x + 2)/9.
- Thus y = (x + 1)[e^(3x)(3x + 2)/9 + C]. This expression also extends smoothly across −1 and satisfies the original equation there.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Divide by x + 1. P = −1/(x + 1), so choose IF 1/(x + 1).
- [1 mark] Then [y/(x + 1)]′ = e^(3x)(x + 1).
- [1 mark] By parts, ∫e^(3x)(x + 1)dx = e^(3x)[(x + 1)/3 − 1/9] = e^(3x)(3x + 2)/9.
- [0.5 marks] Thus y = (x + 1)[e^(3x)(3x + 2)/9 + C]. This expression also extends smoothly across −1 and satisfies the original equation there.
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