Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #58
Solve (arctan y − x)dy = (1 + y²)dx.
✅ Step-by-Step Solution
Final Answer: x = arctan y − 1 + Ce^(−arctan y), as a function of y.
- Write dx/dy + x/(1 + y²) = arctan y/(1 + y²).
- Use IF e^(arctan y). With t = arctan y, xeᵗ = ∫teᵗdt + C.
- Integration by parts gives ∫teᵗdt = (t − 1)eᵗ.
- Thus x = arctan y − 1 + Ce^(−arctan y). For a solution graph y(x), restrict to locally invertible branches.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write dx/dy + x/(1 + y²) = arctan y/(1 + y²).
- [1 mark] Use IF e^(arctan y). With t = arctan y, xeᵗ = ∫teᵗdt + C.
- [1 mark] Integration by parts gives ∫teᵗdt = (t − 1)eᵗ.
- [0.5 marks] Thus x = arctan y − 1 + Ce^(−arctan y). For a solution graph y(x), restrict to locally invertible branches.
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