Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #56
Solve (1 + x²)y′ + y = e^(arctan x).
✅ Step-by-Step Solution
Final Answer: y = ½e^(arctan x) + Ce^(−arctan x).
- Divide by 1 + x². Here P = 1/(1 + x²), IF = e^(arctan x).
- Then ye^(arctan x) = ∫e^(2arctan x)/(1 + x²)dx + C.
- Put t = arctan x; the integral is ∫e^(2t)dt = e^(2t)/2.
- Thus y = ½e^(arctan x) + Ce^(−arctan x), for all real x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Divide by 1 + x². Here P = 1/(1 + x²), IF = e^(arctan x).
- [1 mark] Then ye^(arctan x) = ∫e^(2arctan x)/(1 + x²)dx + C.
- [1 mark] Put t = arctan x; the integral is ∫e^(2t)dt = e^(2t)/2.
- [0.5 marks] Thus y = ½e^(arctan x) + Ce^(−arctan x), for all real x.
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